They did until 1925, after which it was renamed into the All-Union Communist Party (and later the Communist Party of the Soviet Union). It had a dual role as both the overarching party of the whole Union as well as the party of the RSFSR.

Basically, the USSR was originally founded with the intention that all the member republics were on equal standing: in reality, Russia was always dominant to some extent and after Lenin's death and Trotsky's position weakening, Stalin centralised it further. (I have a video on more or less this topic in my profile, if I may interest you :p)

As part of that process, I think they just didn't bother having a separate Russian party when everyone knew they were at the top anyway, so they just renamed their Russian CP and integrated all the other parties under it. In fact, they directly said in the Fourteenth Congress (where the name change was decided on) that the Russian role in the union was "self-evident" and that having them separate would just create overlapping institutions.