While traditional Chinese sources and antiquated works on the Tang would have you believe in the supremacy of the Tang militarily, politically, economically, and culturally vis-a-vis its neighbours throughout history, the truth is somewhat murkier. The Tang dynasty was only one major player among many other notable, powerful, and influential states that dominated the international landscape at the time. You had the Korean states, the Uighurs, the Turkic confederations, Nanzhao, the Tibetans, and even the Arabs at times all competing with the Tang. The Chinese tributary system was how the Tang conducted international diplomacy, but it was far from the only way to conduct international diplomacy and the Tang were far from the only influence on their neighbours. Pragmatically and logically, Tang China's neighbours took what they found beneficial from the Tang, whether that be material goods or merely ideas and thoughts, and threw away the rest. I felt like this was important to state because too often, students are taught in school a very antiquated and inaccurate model of relations within the region. I know in my world history classes, we didn't even cover other countries in the region during our studies of Imperial China, and certainly not the Tibetans.

The Tang and the Tibetans arose around roughly the same time period. While the Tang dynasty was unifying and strengthening its hold over the country and neighbouring regions, the Tibetans were also expanding their military power and sway within the region. The focus of the early Tang emperors was frankly, not on the great power to the West, but towards the more historic foes of Chinese dynasties such as Korea and the Turks. Logically, this would make sense: during the reigns of Taizong and Gaozong, Tibet was not an immediate threat to the Chinese empire. Hell, it didn't even border the Tang. While Tibet would demand (and eventually receive) a Chinese princess, both Tibet and the Tang dynasty had other more pressing issues to attend to that would leave comparatively little conflict between the two in these early decades. Chinese records show that the Tang considered Tibet a tributary at the time (though Tibetan sources disagree.)

However, the peace between two rapidly expanding empires was unlikely to last, and following the fall of the Tuyuhun in the 60s, the two major empires, at long last, were now bordering each other. Over the next few decades, the Tang and Tibet would compete on all fronts militarily, which led to Tibetan supremacy in the Tarim Basin temporarily in the 70s. While the Tang would return in full force to the Basin by the 90s, the point is: relations between the Tibetans and Tang would reach a low-point in comparison to the relatively idyllic times of the decades prior. The early 8th century would feature much of the same, though the Tang dynasty would reach what is considered the zenith of the Tang dynasty militarily. During the reign of Emperor Xuanzong, the Tang dynasty would fight on all fronts to expand their burgeoning empire. In a treaty in 732 between the Tang and the Tibetans, the Tang would refer to the two as "Uncle and Nephew," highlighting the advantageous position the Tang found themselves in and thus confirming the relative superiority that the Tang felt towards the Tibetans.

However, following the An Lushan Rebellion, the Tang dynasty was severely weakened. After seven years of discord and civil war in Tang China, the empire was no longer focused on dominating the region and had turned to prioritising basic survival. Tibet would overrun the Tang in the Tarim Basin and its control of trade flowing through the region. It is in this period that Tibet managed to occupy Chang'an and briefly place a puppet emperor on the throne.

In the 80s, we see a new treaty be signed between the two powers. In it, Tibet "acknowledged" the Uncle-Nephew relationship between the two states, but also used it as fuel for their claim to equal standing with the Tang. While saying that China now viewed Tibet as a full equal would be wrong, it is true that they treated the Tibetan epmire more respectfully as a result.

There would be one last major treaty signed between the Tibet and Tang which enlightens us as to Tang perspectives on Tibet. In the last treaty, the Tang dropped usage of the Uncle-Nephew term and also state the following:

As for the whole control of China, Tang is the sovereign, and as for the whole region of the western frontiers, Great Tibet is the ruler.'

This last and final treaty signed between the two nations reveals the changing nature of the relationship between the Tang and the Tibet. Despite previous attempts at asserting the superiority of China, Tibet would be seen as an equal in the late stages of Tang-Tibet foreign relations, reflecting the complexities of international diplomacy and the world order at the time, which traditional understandings of the tributary system and Sinocentrism would have failed to cover. Hopefully, that answers your question! Definitely recommend reading the source by Pan Yihong if you're interested in some further information since it's short and she goes into more depth about the specifics of the treaties (and also some of the treaties that I skipped.)

Sources:

Ideas concerning Diplomacy and Foreign Policy under the Tang Emperors Gaozu and Taizong. By Wang, Zhenping. Asia Major, third series, 22, no. 1, 2009.

Tang China in Multi-Polar Asia: A History of Diplomacy and War. By Wang, Zhenping. University of Hawaii Press, 2013.

The Sino-Tibetan Treaties in the Tang Dynasty. By Pan, Yihong. T'oung Pao, second series, 78, no. 1/3, 1992.

Tibet in Tang's Grand Strategy. By Twitchett, Denis, in Warfare in Chinese History. Edited by van de Ven, Hans. Brill, 2000.