So I can't find anything on these and wikipedia provides little to no information or is even contradicting itself.
I understand that Iran in that period was a mixture of Persian and old Parthian nobility. Many authors refer to Parthians as being divided into seven clans: Who were they? Where the clans like cadet branches of the Arsacids? What happened to them after the Islamic conquest? Where the Sassanian period Parthian clans the same with the Arsacid period clans? Or was this all made up in the Sassanian period for legitimacy of their rule. And lastly: How were they different from other nobility?
Heres a very cursory answer until someone more informed comes along.
For what happened to the Parthian nobility, they pretty thoroughly assimilated. The Parthian dynasty was in power for close to 300 years and made their capital at Ctsesiphon just outside the area that was the heartland of the Achaemenid empire and closer to the Seleucid capital. This reflects how the Parthians really fell into step with the previous dynasties and co opted this tradition to legitimise themselves. When the last parthian king died, the parthian nobles continued their traditional role in the state and court. I can't recall exact names, but you can detect the same noble families' names in the records in both the Parthian and Sassanian periods, showing that some of the more powerful families endured.
I can't speak at all to Parthian clans, this was never brought up. I would speculate that the clans/clan structure would be almost totally, if not totally gone well before the islamic conquest since the Parthians were very good at assimilating with persian traditions. Don't take my word on this though, its my own guess based on my own knowledge of the subject, but I have no direct source.