  Recently, Andrews introduced separable integer partition classes and analyzed
some well-known theorems. In this paper, we investigate partitions with parts
separated by parity introduced by Andrews with the aid of separable integer
partition classes with modulus $2$. We also extend separable integer partition
classes with modulus $1$ to overpartitions, called separable overpartition
classes. We study overpartitions and the overpartition analogue of
Rogers-Ramanujan identities, which are separable overpartition classes.
