Not at all. Incan society had a caste system, nobles, royalty, etc. Nobles owned estates and laborers. Workers did not own the means of production.
A more fair comparison would be to North Korea. They're still old-school communist, and really not doing well.
The concept of nobility implies hereditary transfer of privileges, which was not the case in Soviet system. It was certainly easier for children of Party bureaucrats to get their own spot, but it was not guaranteed to them, they could lose it at any time, and "commoners" could get into the ranks (and it was not something exceptional).
So Party was the ruling elite class, but it was not an estate/caste, as nobility was in feudalism.