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Sang, Shi and Yee, in 2020, found overpartition analogs of Andrews' resultsinvolving parity in Rogers-Ramanujan-Gordon identities. Their result partiallyanswered an open question of Andrews'. The open question was to involve parityin overpartition identities. We extend Sang, Shi, and Yee's work to arbitrarymoduli, and also provide a missing case in their identities. We also unifyproofs of Rogers-Ramanujan-Gordon identities for overpartitions due to Lovejoyand Chen et.al.; Sang, Shi, and Yee's results; and ours. Although verificationtype proofs are given for brevity, a construction of series as solutions offunctional equations between partition generating functions is sketched.

Modulo $d$ extension of parity results in Rogers-Ramanujan-Gordon type overpartition identities