  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 $\overline{C}_{k,i}(n)$ of $(k,i)$-singular overpartitions of $n$. In
particular, we give a sufficient condition on even values of $k$ so that the
values of $\overline{C}_{4k,k}(n)$ are almost always even. Furthermore, we show
that for odd values of $k \leq 23$, $k\neq 19$, certain subsequences of
$\overline{C}_{4k,k}(n)$ are almost always even.
