When did horseback riding start? Are Doosheh cave petroglyphs authentic and correctly dated?

There seems to be something going on of the quite unreliable kind. The generally accepted 'history' and dates of horse domestication and riding practices are quite at odds with the dates for cave carvings and paintings from Iran sometimes given for 'riders' so early, and so early in that region.

It seems that the rock-art in the غار دوشه Doosheh cave cannot be dated to so early, and apparently wasn't dated reliably, or really: dated at all, as almost all such sites in Iran in general, and that the domesticated horse itself made its appearance in Iran much later, matching more the first millennium BCE time as conservatively assumed. Without any horses present archaeologically, cave art of this kind has to be younger. Iranian tourism websites don't mention any references for their claims, but stick at least with 4500 BC as the time frame. That would be very early, perhaps very much too early indeed, but not that much out of the ordinary like this '15000 years' claim.

An exact dating for what's found at Dousheh (spelling used in English Wikipedia for Rock Art in Iran) is not available to me, but the Tehran Times ("Straight Truth" [!]) reuses the rider given in the question's picture and states in 2021 for another find in Lorestan province:

“The petroglyphs, which bear carved symbols and figures in the two colors of black and ocher, were discovered during an architectural survey conducted in Chegeni county of Lorestan province,” the provincial tourism chief, Seyyed Amin Qasemi, announced on Monday.

“The drawings include animal, human, and plant motifs as well as scenes of hunting and horseback riding,” the official said. […]

International experts Jan Brouwer and Gus van Veen have examined the Teymareh site estimating its carvings were made 40,000-4,000 years ago.

— "Ancient petroglyphs discovered in western Iran", Tehran Times, February 16, 2021.

And this is a problem. It seems we do not have any exact dates for the very pictures of interest: those of the horseback riders in particular. Dating some of these petroglyphs or cave paintings, and in this big range, doesn't give us much confidence for the ones showing mounted men. Some art is superimposed on or side by side with others of different ages. This seems visible even in the picture used within the question, with figures apparently 'in the background' appearing much more faded? Moreover, further information from scientific journals within this speciality field should give some further pause.

For example: This journal has a pretty authoritative name on the subject matter and declares:

Detailed researches on the chronology of petroglyphs have not been carried out in Iran, so we cannot suggest any date…"

— Mousa Sabzi & Esmail Hemati Azandaryani: "Bichoun: newfound rock art at Boroujerd, Lorestan Province, western Iran", Rock Art Research, Volume 34, Number 1, 2017.

With exact dating on this very issue not progressing very far recently:

The purpose of the creators of these motifs still is vague and unknown to us. Regarding the dating of these motifs, it should be noted that, due to the lack of laboratory equipment, no absolute dating has been done so far, but relative dating can be provided for them, which can have dated them from the Neolithic to the contemporary periods.

— Mousa Sabzi & Esmail Hemati Azandaryani: "Investigation and Analysis in the Boroujerd’s Petroglyphs, Lorestan Province", بررسی و تحلیل سنگنگارههای بروجرد، استان لرستان (Pazhohesh-Ha-Ye Bastanshenasi Iran), Volume 10, No 25, pp91–112 2020. doi:10.22084/nbsh.2019.17643.1844

And

[…] with respect to the horse-rider depictions, it should be noted that horses were brought into the Iranian Plateau for the first time in the 2nd millennium BCE by Aryan immigrants.

— Ebrahim Karimi: "The Rock Paintings of Kuh-e-Donbeh in Esfahan, Central Iran", Arts, 2014, 3, pp118–134; doi:10.3390/arts3010118

Comparing the style evidenced by the rider shown to us in question, one starts to wonder comparing it to this, written by Iranian scientists:

Panel 2. On this panel, we find the human motifs and ibexes. But they are not contemporaneous because the shades of patina are not the same. It seems that anthropomorphic images are younger because the color is brighter (Fig. 15). The technique of engraving is coarser than Aps-e Goalm images.

Panel 3. This is the richest panel containing different animal and anthropomorphic images. Among the animals, horses, Bactrian camels, ibexes, goats, zebus, and dogs are visible. Two human motifs are shown in this panel: one stand up in the front of a zebu and another

These motifs represent tamed domesticated horses with riders. Comparable motifs also have been previously identied at Mir Malas Cave in Luristan dated back to the Iron Age (McBurney, 1969). Such dating could be applicable to the Qasr-e Qand petroglyphs too. […]

Panel 1. This panel contains anthropomorphic and zoomorphic motifs (Fig. 14). One motif is very useful for dating of images. Here, a man is controlling the animal (probably a horse) by taking the muzzle. Some ibexes are shown in the left side of the panel beside some unidentiable creatures. The horse image is attested at an early period in the mobile art as well as on the rock art of Inner Asia and Iranian Plateau, from Altai to Baluchistan. As Francfort pointed out,

“…after having been hunted during a long period, the horse became the main mount and domestic animal very familiar to the nomadic peoples from the 1st millennium BC to our days…"

In Iran, the introduction of the domesticated horse corresponding to the Indo-Aryan peoples dates back to the Iron Age. Archaeological evidence afrms the presence of the domesticated horse in northern Iran at the end of the 2nd millennium BC. […]

— R. Shirazi & M. Soltani: "The Evidence Of Rock Art in Iranian Makran: Aps-E Goalm and Kouhbodan-E Jor Petroglyphs, Qasr-E Qand", Archaeology Ethnology & Anthropology of Eurasia 43/2 (2015) pp23–33. doi:10.1016/j.aeae.2015.09.003

The paragraphs above are shortened, rearranged, and the article has some additional pictures (the one shown here declared as a camelid, though) that to my eyes make it quite unlikely that the dating of 15000 BP or BCE are trustworthy. Coming back to the site giving this extremely early date of 15000, it also says with already a clearly propagandistic effect gathering title:

"Older Than The Egyptian Civilization"

The first tamed animals, such as the dog, the horse, the sheep and the goat, are native to Iran, and the discovery of the bones of these animals in the ancient caves of the land show that Iranians were the first people to tame animals. In Doosheh cave, near the city of Khorramabad, considerable pictures dating back to 15,000 BC have been found which show men riding horses and holding the animals' reins. The oldest evidence showing the use of horses in Mesopotamia and Egypt, however, date back to only 4,000 years ago. Also, according to researches carried out by Professor Morris Damas in his book, "The History of Industry and Invention", it was the Arian race that first utilized wheeled carriages pulled by the horses.

My trust level in this kind of presentation is not the highest.

And yet, this is the source referenced in the Book "A Concise History of Iran: From the Early Period to the Present Time" on page 21, footnote 9. And it is therefore identical to what in the question is called "Another mention…". Thus, what first seemed like two sources is really just one, and a fishy one, giving itself no further references or even clues how 'that early date' came to be.

The Persian language Wikipedia entry on this cave claims 'Dushe cave to be 8000 years old' and gives a picture that is captioned "around the eighth millennium". —
This is not in accordance with the only two sources mentioned on that Wikipedia page. The second one hamshahrionline.ir says:

go to a painting gallery that is more than 5,000 years old. Khorramabad Dusheh Cave has 110 carvings carved on the rocks by early humans. […]
What is certain and can be deduced from these motifs is that the history of murals and paintings in Iran dates back to at least the Neolithic period, the New Stone Age, ie around 4500 BC.

The first reference given on Persian Wikipedia is from Iranian Students News Agency (ISNA) is actually saying:

Dusheh Cave is the oldest gallery in Iran, which is 8,000 years old.
According to the Iranian Students News Agency (ISNA), Lorestan region is one of the historical caves of Ghar Dosheh province, which is located in the village of Karshorab in the Chegni district of Khorramabad.
The walls of this cave contain numerous and interesting historical paintings and inscriptions [emphasis added, LLC]

This field of study for rock-art is young in Iran, not widely accepted there, prone to certain confounders in dating generally, … and the problems continue.

One of the problems is indeed said to be directly related to political sanctions, giving us here just further problems:

In 2008, archaeology experts from the Netherlands visited the area with Dr Naserifard and said a cluster of drawings made using cups could be more than 40,000 years old.

One of the most striking pieces is a 4,000-year-old engraving of an ibex deer, complete with long curled horns, at the top of a hill. […]

“But sanctions have deprived us of the technology. We hope we can soon bring this technology to Iran and gain more accurate and scientific information on these engravings.”

Iranian scientists are not allowed to use uranium analysis tools in their research due to US sanctions. The technology acts as a new form of carbon dating which can accurately estimate when something was created.

— Gabriel Samuels: "'World's oldest rock drawings' uncovered in Iran by archaeologist", The Independent, Monday 12 December 2016.

This would explain in part the profound lack of what many here would consider a requirement for dating things:

Compared with neighbouring countries, the history of rock art studies in Iran has been short. Such studies extend to 60 years ago when an Italian geological team reported some petroglyphs in Iranian Baluchistan (Deassau 1960). For a long time, information concerning rock art in Iran remained confined to this work and a few other sites like the Mirmalas and Ducheh paintings in the west (Izadpanah 1969; McBurney 1969).

Despite this wealth of rock art, is not accepted as part of the archaeological discipline in Iran. A main reason for such rejection relates to the chronology of rock art. Without identifying the age of rock art, this evidence is of no help to the archaeologist, because it can be linked to archaeological constructs only by the factor of age (Bednarik 2002). None of the methods of direct dating have been applied to Iranian rock art, and traditional methods resulted in relative age estimates only. […]

The most common motif type of Iran’s rock art are zoomorphs and particularly ‘ibex’ motifs characterised by exaggerated ‘horns’. This animal is often shown with ‘horse riders’ in apparent hunting scenes. Regarding the identification of the domesticated horse from the Iron Age in the first millennium BCE onwards and its employment to draw vehicles in the Historic-Islamic era, the rock art of presumed hunting scenes with ‘horse riders’ cannot be of pre-Historic time and is thought to be at most of the first millennium BCE (Mohammadi Ghasrian 2017). […]

An understanding of the antiquity of rock art sites is considered as a main methodological limitation of rock art studies in Iran. Here, I have presented some new chronological observations. Based on them, around 31 sites of Iran’s rock art are proposed to be of the Historic to Islamic periods and not pre-Historic. My chronological remarks have their shortcomings too: first, the Historic-Islamic period is a long time span, although this does serve to separate pre-Historic rock art from subsequent traditions. Second, some of Iran’s rock art, like that at Timreh, Houmyan, Saravan, Makaran and elsewhere, occurs as site complexes (Fig. 1), characterised by ‘geometrics’, ‘horse riders’ and further motifs. At site complexes such as Timreh with 30 000 or so petroglyphs these would not all be contempora- neous and of the Historic-Islamic period, obviously. Third, the mentioned geometric motifs used as an age indicator for dating rock art to Historic-Islamic period may derive from the pre-Historic period.

— Sirvan Mohammadi Ghasrian: "Antiquity of Iran’s rock art: pre-History or Historic-Islamic time?", Rock Art Research, Vol 37, No 1, 2020. (academia.edu)

These probably disconcerting news for the Doucheh cave as such and the connected '15000 years' claim may be complemented with a firmly contradicting maximum 'earliest date', from another geographical area, for "the main question", or the 'earliest horseback riding—much earlier than previously thought' (a date still to take with some very big grains of salt):

Analysis of horse teeth from Ukraine proves that horse riding began 6000 years ago, much earlier than had been supposed. The oinnovation affected the dispersion of culture and language.

— David Anthony & Dorcas R Brown: "The Origin of Horseback Riding", Scientific American, Vol. 265, No. 6, (December 1991), pp. 94-101. jstor

Similarly:

It first occurs in the Ukraine, USSR, at about 4000 BC. Soon thereafter, mobility became a cultural advantage that transformed Eurasian societies. Riding now appears to be the first major innovation in land transport, pre-dating the wheel.

— David W. Anthony & Dorcas R. Brown: "The origins of horseback riding", Antiquity, Volume 65, Issue 246, March 1991, pp22–38 doi:/10.1017/S0003598X00079278

The grains of salt were needed, because this dating itself is also a matter of scientific controversy:

Attempts by some scientists (Anthony 1986; Zaibert, et al. 1990, etc.) to interpret bone articles from Eneolithic sites in Eastern Europe and northern Kazakhstan (i.e., Derievka, Tjubek, Botai, etc.) as cheekpieces, and accordingly, to extrapolate the development of horseback riding from this information, is not absolutely convincing for a number of archaeological reasons (Bokovenko 1997). In addition, their theories have been developed in the absence of reliable osteological data (Kosintsev 1999; Levine 1999).

It was only at the end of the Bronze Age that a sporadic development in the steppe cultures occurred in which horseback riding was mastered–and this was probably by shepherds. It is during this time that finds of bone and rod-shaped cheekpieces with various modifications are noted, as well as being illustrated in numerous rock drawings (Bokovenko 1979; 1986).

— N. A. Bokovenko: "The Origins of Horse riding and the Development of Ancient Central Asian Nomadic Riding Harnesses", Kurgans, Ritual Sites, and Settlements: Eurasian Bronze and Iron Age, BAR International Series 890. 2000. pp304–310. (pdf)

Thus the currently conservative consensus position on these developments — first horse raising, then quite a bit later riding — and dates reads:

Despite its geographically dispersed archaeological landscape, multiregional interaction is a defining feature of the Central Asian Bronze Age. Dynamic transregional connections and exchange in the region is shown most clearly through its second millennium BC metallurgical inventory, horse riding and chariot technology, and geographic transport of crops between China and Southwest Asia (Sherratt 2006). Archeological discoveries made over the past ten years, however, demonstrate that a regional web of interaction and technology transfer was in place earlier on, during the third millennium BC (Jones et al. 2011; Frachetti 2012; Spengler et al. 2014). These discoveries are the result of a growing number of archaeometric dates, settlement excavations with multiperiod stratified cultural deposits, site survey and mapping, and scientific studies of diet and exchange. […]

The Early Bronze Age in northern Central Asia dates from approximately 3200 to 2400 BC […] Northern Eurasia is home to pastoral groups of the Botai and Tersek cultures (ca. 3600–2500 BC), with their sites spanning the Tobol and Irtysh river basins or northern Kazakhstan and Siberia (Kislenko and Tatarintseva 1999). Their subsistence economy was focused on horse rearing, milking, and meat consumption (Outram et al. 2009). Domestic economies of the Early Bronze Age, although showing that pastoral economies were present in the region by the early third millennium BC, more so highlight the varied aspects of regional herd compositions and subsistence forms, and possibly limited regional contact, for this early period.

— Paula Doumani Dupuy: "Bronze Age Central Asia", Oxford Handbooks Online: Archaeology, Archaeology of Central Asia, 2016. DOI: 10.1093/oxfordhb/9780199935413.013.15