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.