  Let $pod_{\ell}(n)$ be the number of $\ell$-regular partitions of $n$ with
distinct odd parts. In this article, prove that for any positive integer $k$,
the set of non-negative integers $n$ for which $pod_{\ell}(n)\equiv 0
\pmod{p^{k}}$ has density one under certain conditions on $\ell$ and $p$. For
$p \in \{3,5,7\}$, we also exhibit multiplicative identities for $pod_{p}(n)$
modulo $p.$
