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The $(k,i)$-singular overpartitions, combinatorial objects introduced byAndrews in 2015, are known to satisfy Ramanujan-type congruences modulo anypower of prime coprime to $6k$. In this paper we consider the parity of thenumber $\overline{C}_{k,i}(n)$ of $(k,i)$-singular overpartitions of $n$. Inparticular, we give a sufficient condition on even values of $k$ so that thevalues of $\overline{C}_{4k,k}(n)$ are almost always even. Furthermore, we showthat for odd values of $k \leq 23$, $k\neq 19$, certain subsequences of$\overline{C}_{4k,k}(n)$ are almost always even.

Density results for the parity of $(4k,k)$-singular overpartitions