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

The $(k,i)$-singular overpartitions, combinatorial objects introduced by Andrews in 2015, are known to satisfy Ramanujan-type congruences modulo any power of prime coprime to $6k$. In this paper we consider the parity of the number ${\bar{C}}_{k,i}(n)$ of $(k,i)$-singular overpartitions of $n$. In particular, we give a sufficient condition on even values of $k$ so that the values of ${\bar{C}}_{4k,k}(n)$ are almost always even. Furthermore, we show that for odd values of $k\le 23$, $k\mathrm{\ne }19$, certain subsequences of ${\bar{C}}_{4k,k}(n)$ are almost always even.