PED partitions are partitions with even parts distinct while odd parts are unrestricted. Similarly, POD partitions have distinct odd parts while even parts are unrestricted. In [16] several recurrence relations for the number of PED partitions of n are proved analytically. They are similar to the recurrence relation for the number of partitions of n given by Euler's pentagonal number theorem. We provide combinatorial proofs for all of these theorems and also for the pentagonal number theorem for PED partitions proved analytically in [11], which motivated the theorems in [16]. Moreover, we prove combinatorially a recurrence for POD partitions given in [7], Beck-type identities involving PED and POD partitions, and several other results about PED and POD partitions.