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We continue the study of the $(a,b,m)$-copartition function$\mathrm{cp}_{a,b,m}(n)$, which arose as a combinatorial generalization ofAndrews' partitions with even parts below odd parts. The generating function of$\mathrm{cp}_{a,b,m}(n)$ has a nice representation as an infinite product. Inthis paper, we focus on the parity of $\mathrm{cp}_{a,b,m}(n)$. As with theordinary partition function, it is difficult to show positive density of eithereven or odd values of $\mathrm{cp}_{a,b,m}(n)$ for arbitrary $a, b$, and $m$.However, we find specific cases of $a,b,m$ such that $\mathrm{cp}_{a,b,m}(n)$is even with density 1. Additionally, we show that the sequence$\{\mathrm{cp}_{a,m-a,m}(n)\}_{n=0}^\infty$ takes both even and odd valuesinfinitely often.

On the parity of the number of $(a,b,m)$-copartitions of $n$