My apologies for the late reply. This turned out to be a more complex and interesting question than I had anticipated. I had hoped to be able to unearth and make use of some Chinese primary sources, but it looks like the situation is hopelessly muddled and I'd have to read some Chinese dissertations (in Chinese) to straighten things out, and I don't read Chinese, so... this answer will essentially be entirely based on the chapter "Zoroastrianism in the Far East" in "The Wiley-Blackwell Companion to Zoroastrianism" (2015).

Zoroastrianism, or "heaven worship" has a history that precedes the flight of Sasanian nobles to the Tang Dynasty. In fact, it is roughly contemporaneous with the emergence of the Sasanian dominion, beginning during the 4th century or so, as Sogdians begin moving into the territory of the Wei and Jin. Some fragments in Sogdian of a religious nature survive (the Asham Vohu, a key prayer, and a Zoroaster asks Ahura Mazda questions-style document), but there are no preserved Chinese books on Zoroastrianism (unlike the situation with Manichaeism and Eastern Christianity). This may be related to the fact that the Tang Dynasty (618-907) prohibited the Chinese from converting to Zoroastrianism.

Anyway, to give a direct answer to your question: According to Takeshi Aoki, the author of the chapter, the sinicization of the Zoroastrians began in 5 Dynasties/10 Kingdoms period in the 10th century, and was a continuous process lasting until the present day. Moreover, the last member of the House of Sasan to be recorded in Chinese records is a Khosrow who died in 728 AD. In the mid-7th century Peroz III had pleaded to the Gaozong emperor for an army to retake the lost lands, and the emperor obliged. For some decades Peroz and his son Naseh did manage to retake what was probably Sistan. Peroz died in 707 AD; Narseh invaded Tokharistan with a Tang army and reigned there for twenty years until he died in 727 AD. That is, one year before the aforementioned "Khosrow" dies.

Therefore, it looks like the Chinese records of the Sasanian royal family is tied to their military adventures, which seem to have died with Narseh. On the other hand, the story of Zoroastrianism itself in China lived beyond that. For example, one general by the name of Roχšan, a half-Sogdian Zorostrian (per the arguments in Rong 2006, which I can't read because it's in Chinese) started a rebellion in the mid-8th century, that tore the empire apart.

He is better known by his sinicized name, An Lushan.