Pairs of partitions without repeated odd parts

We prove two identities related to overpartition pairs. One of them gives a generalization of an identity due to Lovejoy, which was used in a joint work by Bringmann and Lovejoy to derive congruences for overpartition pairs. We apply our two identities on pairs of partitions where each partition has no repeated odd parts. We also present three partition statistics that give combinatorial explanations to a congruence modulo 3 satisfied by these partition pairs.