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Ballistae and other ancient pieces of "artillery" are siege engines. Their primary purpose is to provide fire support within the context of laying siege to a town or fortress; the heavy bolts could lay waste to wooden fortifications (especially the kind of light mobile protection against archers). Siege weapons are heavy, very slow to move, and have a low firing rate. As such, it was extremely rare for such weapons to be used for anything else than sieges.
There are very few documented uses of artillery in field battles; the most ancient is the battle of Jaxartes in which Alexander the Great ordered the use of some siege engines to clear the opposite river bank, from which archers were giving trouble to crossing Macedonian troops. The involved weapons are often designated as "catapults" but were most probably firing bolts. The first documented use of rock-lobbing weapons as field artillery is much later, under command of Mongol general Subutai, who used such engines to, again, clear a river bank from crossbowmen who were threatening crossing troops (at the battle of Mohi, in 1241). In all these cases, the same pattern is present: some army is involved in a pitched battle; it also has some catapults available, meant for an ulterior siege of some town; a brilliant general notices a situation where the catapults could be of some use, and proceeds to do it. This is always exceptional, and reported as such. Crucially, no army sets out with catapults designed for field usage.
We must make an exception for light one-man catapults like the Roman scorpio, which really is a crossbow on a tripod. Up to the Renaissance, the role of "field artillery" (as it is understood nowadays) was fulfilled by light infantry with javelins, slings and bows (and, in the Roman case, scorpios). These specialized troops would move fast across the battlefield, spread out, and harass heavy infantry or cavalry. Cannons finally replaced them, when their increased firing range began to make up for their lack of mobility.
Thus, an ancient army would not have included ballistae or equivalent unless it planned siege warfare. But siege warfare requires more than siege engines; it involves earthworks, logistics, architecture specialists... in one word, it needs engineers. Romans were famed for their expertise in that area; the same does not apply to other armies of that time. In particular, the "Persians" from the days of Septimius Severus were actually Parthians, originally a nomadic people from central Asia. The core of their army would be mounted archers. This is, again, a common strategic historical pattern: siege warfare is something that is learned from experience, after encountering an opponent who knows a good deal more on the subject.
So the lack of artillery in the Persian armies is not really a problem of building the weapons themselves; it more is a question of warfare doctrine. Persians would not have much use of catapults alone, until they knew how to conduct sieges, knowledge that they acquired, indeed, from the Romans.
Upvote:1
The main impediment for a nation like the Persians is the relative lack of a standing army. The Persians were a feudal state. In order to go to war "big time" with the Romans, the King of Kings would have to convince his district "kings" to send contingents to join his own household troops. If the KofK's was a weak one, nobody would show up and the Romans could run rampant, as shown in the several cases of Roman armies sacking the capital, Ctesiphon.
These feudal contingents were mostly cavalry, notoriously reluctant to do heavy lifting, and were dispersed themselves between wars. Engineering requires some training and discovery, and much crafting and practice. Without a standing army, this is nearly impossible to keep going. Even Rome, with a much stronger tradition of work, limited their sieges to blockades such as the 10 year siege of Veii, or the case of Nola in the Social War. After the growth of the long term army under Caesar in Gaul, and in the Empire the amount of engineering action grows dramatically. The professional armies of the Empire had the coherency and time to develop these skills, and the work ethic to get the job done in the field.
Upvote:3
Artillery was a big man's trade as much 2,000 years ago as it is now, or was in Napoleon's age. Whether it is lugging the shells around today, or winding the windlass then, strength is vital for a better rate of fire, and strength derives from physical size.
Further, in addition to a team of gorillas to man the piece, a gun layer (or two, as a spare) is required for each piece to properly range it so that the fire hits the enemy instead of harmlessly striking the ground before or behind them. This crew member would also have been the engineer responsible for maintenance and repair of the piece.
In order to have a sizeable artillery corps a nation must have a culture where it is acceptable for a big brawny man to study mathematics and architecture and tinkering instead of melee combat. It must also have educational institutions capable of properly training such individuals.
I suspect that the Persians simply didn't expend enough effort in peacetime to build the necessary infrastructure, making them permanently dependent on captured Romans and Greeks in wartime.
Update: According to this source (The Roman Army at War: 100 BC - AD 200, Goldsworthy, page 61) regarding the Parthians:
Although the king controlled the army, it was recruited on a feudal basis from the noble families and their retainers. ... Contingents served together under their own leaders and in most instances seem to have been loyal to them, following them into exile The greater nobles often possessed both the ability and the will to challenge the king for the throne or support one of his relatives in doing so.
References by the above: