Update: I misread the question. I thought that the poster was asking for when the height of the Silk Road was. There certainly were three periods of intense Silk Road activity. They correspond to periods of relative stability in Asia, which were interspersed by much longer periods of instability, warfare, and nomadic invasions.

The first period was in the late classical era c. 100 B.C.-250 A.D. It commenced after the Han Dynasty had successfully expanded westward and established contacts with the Parthians, in 124 B.C. It lasted until about 220 A.D., when the Han and Roman empires began to fracture.

The second period corresponds to the T'ang Dynasty of China, c. 7th-9th centuries. The third period was Pax Mongolica.

Old Answer:

I've not been able to define when the height of the Silk Road was. The initial Silk Road period is characterized as a sort of golden age by which other periods were measured. It was certainly the most idyllic in terms of peaceful economic activity. This probably wasn't the period of greatest activity, though. Imperialism and competition for lucrative Silk Road commerce led to its disruption as early as the 3rd century:

The agressive foreign policy successes of the Chinese and Roman empires ultimately had disastrous consequences. The partial closing of the frontier to trade by both empires, and their destabilization of Central Eurasia by their incessant attacks, resulted in internecine war in the region. The serious decline in Silk Road commerce that followed- observable in the shrinkage of the areal extent of Central Asian cities- may have been one of the causes of the long lasting recession that eventually brought about the collapse of both the Western Roman Empire and the Eastern Han Empire (and its eventual successor the Chin Dynasty), and with them the end of Classical Civilization. Beckwith, Christopher. Empires of the Silk Road. Princeton University Press, 2009. pg. 92

The Medieval Sogdians are considered the height of Medieval Silk Road civilization in Central Asia. Sogdian Civilization rose and fell with the T'ang dynasty in China, from the 7th to 9th centuries. Both of these, along with the Abbasid Caliphate of Baghdad fractured in the 9th century, and collapsed in the early 10th century. Asia became beset by economic troubles and nomadic invasions. The instability in Asia was not remedied until Pax Mongolica.

I'd guess that it was the T'ang period which saw the most activity because China was so focused on Silk Road commerce and Central Asian foreign policy. The T'ang Empire stretched into Mongolia, and they successfully gained control of the Eastern Turks in Iran. Baghdad was the largest city in the world in the 9th century because of its location at the junction of Silk Road and Mesopotamian trade routes. If not the greatest in volume, the Sogdians represent the height of the Silk Road civilization.

The world population began to rapidly increase around 1000 A.D., so it was much greater in the Mongolian period. Global commerce increased during Pax Mongolica. However, maritime routes between the west and China had developed a lot in the preceeding centuries. Central Asia was no longer the only way to transport goods across Eurasia. However, if you consider the definition of the Silk Road to be all trade between East and West, then it was probably greatest during Pax Mongolica, in the 14th century.

Short Answer

There is support among some historians for identifying three periods of major activity along the Silk Road. Some list only two, others four (by dividing one of the three into two halves). But the three often cited coincide with the following dynasties:

1. The Han Dynasty / First Pax Sinica;
2. The T'ang Dynasty / Second Pax Sinica;
3. The Yuan Dynasty / Pax Mongolica.

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Long Answer

Some historians date these Silk Road periods according to the indicated dynastic ages while others are more precise in dating the usage of the Silk Road within those dynastic periods. Specific examples are included below.

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Smithsonian Institute

According to the Smithsonian program book The Silk Road: Connecting People and Cultures by Richard Kurin, as cited on the Smithsonian Institute's web site in 2002, the three conventionally referenced periods are dated:

1. 206* B.C.E to 220 C.E. (Smithsonian lists these, they date the Han Dynasty);
2. 618 C.E. to 907 C.E. (the T'ang Dynasty);
3. 13th and 14th Centuries C.E. (Mongolian control).

... with the additional consideration of a 4^th Modern period, beginning in the 19^th Century through today.

^{* The beginning of the Han Dynasty in 206 BCE predates the start of the Silk Road in ~139-130 BCE. Refer below to the China Knowledge and Travel China Guide and Wikipedia sections for details on Zhang Qian's mission West, which began the Silk Road story.}

Here is the citation on the Smithsonian's web site:

Conventionally, historians refer to three periods of intense Silk Road trade: 1) from 206 B.C.E. to 220 C.E., between the ancient Chinese Han dynasty and Central Asia, extending to Rome; 2) from about 618 to 907 C.E., between Tang dynasty China and Central Asia, Byzantium, the Arab Umayyad and Abbasid empires, the Sasanian Persian Empire, and India, and coinciding with the expansion of Islam, Buddhism, Assyrian Christianity, Zoroastrianism, Manicheism, and Judaism into Central Asia; and 3) during the 13th and 14th centuries, between China, Central Asia, Persia, India, and early modern Europe, made possible by Mongol control of most of the Silk Road. Some would add a modern Silk Road period, beginning in the 19th century with the "Great Game" — the competition between Russian and British colonial powers for influence over Central Asia — and extending through today. [emphasis added]

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China Knowledge

China Knowledge is "An Encyclopaedia on Chinese History, Literature and Art." Their Silk Road article indicates the following three periods of major usage:

1. The Former and Later Han Dynasties, beginning with the mission of Zhang Qian to Central Asia in 139 B.C.E. and ending with the demise of the Han Dynasty in 220 C.E.;
2. The Reunification of China under the Sui [581-618 B.C.E] and Tang Dynasties [618-907 B.C.E.];
3. Pax Mongolica [~1234-1368 C.E.]

Below are the relevant quotes.

Period 1:

The attested history of the Silk Road begins, from the Chinese side, with the mission of Zhang Qian 張騫 (164-114 BCE) to Central Asia in 139 BCE. ... Not even the turmoil at the end of the Former Han and the creation of the Later Han put a real end to this "early globalization". Ban Chao 班超 (32-102 CE) restored the suzerainty of Eastern Han China over the Western Territories. ... After the demise of the Han dynasty, China was for three centuries (most of the time at least) divided into several states. Even if the Chinese demand for foreign luxury goods was not as great any more as during the Han period, the rulers of the north, like the Former Liang dynasty 前涼 (314-376), did all they could to foster good trade relations with the states and tribes in the northwest. [emphasis added]

Period 2:

The reunification of China under the Sui 隋 (581-618) and Tang dynasties revived the trade routes and gave the impetus for an even more intensive exchange. ... Two events ended this brilliant age of the Silk Road. The first was the rebellion of An Lushan 安祿山 in the mid-8th century that devastated Chang'an and weakened the Tang dynasty financially. The second was the northward expansion of the empire of Tubo 吐蕃 (Tibet) that conquered the access routes along the Gansu Corridor and also part of the Western Territories. [emphasis added]

Period 3:

The Silk Road experienced its last apogee under the rule of the Mongols, when the creation of regular postal stations or caravanserais under the so-called "Pax Mongolica" (in analogy to the Pax Romana) allowed free movement of merchants. ... The Silk Road as a single, unified trade network from China to the Levant was ended after the withdrawal of the Mongols into the steppe and the foundation of the Ming dynasty 明 (1368-1644). [emphasis added]

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Travel China Guide

This Traveler's Guide to China includes an article on the History of the Silk Road which identifies four principle periods of major trade, as follows (the first two below are consolidated into one Han Dynasty period by other historians):

1. The Western Han Dynasty, beginning with Zhang Qian's quest (139 B.C.E.) and ending in 24 C.E.;
2. The Eastern Han Dynasty (25-220 C.E.);
3. The Tang Dynasty (618-907 C.E);
4. The Yuan Dynasty (1271–1368).

Source: History of the Silk Road.

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The Khan Academy

The Khan Academy includes course material on The First Silk Roads which identifies these two periods of major commerce along the Silk Roads:

1. The First Silk Roads Era (50 B.C.E. – 250 C.E.) [Han Dynasty];
2. The Second Silk Roads Era (700–1200 C.E.) [Tang and Song Dynasties]

Here is the citation from this course:

The first major period of Silk Roads trade occurred between c. 50 BCE and 250 CE, when exchanges took place between the Chinese, Indian, Kushan, Iranian, steppe-nomadic, and Mediterranean cultures. A second significant Silk Roads era operated from about 700 to 1200 CE, connecting China, India, Southeast Asia, the Islamic realm, and the Mediterranean into a vast web based on busy overland and maritime trade. [emphasis added]

[There is more detail on these two eras included in the course material, only the summary from one course is cited above.]

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Wikipedia

The Wiki article on the Silk Road requires parsing to find references to periods of major use versus periods of declining or disuse. There are no convenient bullet points identifying these periods. But an examination (including following some additional linked articles) yields these results:

1. The First Pax Sinica (identified as the Han Dynasty) during the Chinese exploration of Central Asia;
2. The Second Pax Sinica (Tang Dynasty);
3. The Mongol Age (1207-1360 C.E.]

Below are some Wiki excerpts:

Period 1:

With the Mediterranean linked to the Fergana Valley, the next step was to open a route across the Tarim Basin and the Hexi Corridor to China Proper. This extension came around 130 BCE, with the embassies of the Han dynasty to Central Asia following the reports of the ambassador Zhang Qian (who was originally sent to obtain an alliance with the Yuezhi against the Xiongnu) ... The Silk Roads' origin lay in the hands of the Chinese ... Past its inception, the Chinese continued to dominate the Silk Roads, a process which was accelerated when "China snatched control of the Silk Road from the Hsiung-nu" and the Chinese general Cheng Ki "installed himself as protector of the Tarim at Wu-lei, situated between Kara Shahr and Kucha." ... The Han army regularly policed the trade route against nomadic bandit forces generally identified as Xiongnu. ... The West Roman Empire, and its demand for sophisticated Asian products, crumbled in the West around the 5th century. [emphasis added]

Period 2:

Although the Silk Road was initially formulated during the reign of Emperor Wu of Han (141–87 BCE), it was reopened by the Tang Empire in 639 when Hou Junji conquered the Western Regions, and remained open for almost four decades. It was closed after the Tibetans captured it in 678, but in 699, during Empress Wu's period, the Silk Road reopened when the Tang reconquered the Four Garrisons of Anxi originally installed in 640, once again connecting China directly to the West for land-based trade... The Tang dynasty established a second Pax Sinica, and the Silk Road reached its golden age, whereby Persian and Sogdian merchants benefited from the commerce between East and West. [emphasis added]

Period 3:

The Mongol expansion throughout the Asian continent from around 1207 to 1360 helped bring political stability and re-established the Silk Road (via Karakorum). ... The fragmentation of the Mongol Empire loosened the political, cultural, and economic unity of the Silk Road. [emphasis added]