  For a positive integer $\ell$, let $b_{\ell}(n)$ denote the number of
$\ell$-regular partitions of a nonnegative integer $n$. Motivated by some
recent conjectures of Keith and Zanello, we establish infinite families of
congruences modulo $2$ for $b_3(n)$ and $b_{21}(n)$. We prove a specific case
of a conjecture of Keith and Zanello on self-similarities of $b_3(n)$ modulo
$2$. We next prove that the series $\sum_{n=0}^{\infty}b_9(2n+1)q^n$ is
lacunary modulo arbitrary powers of $2$. We also prove that the series
$\sum_{n=0}^{\infty}b_9(4n)q^n$ is lacunary modulo $2$.
