  Let $p_k(n)$ denote the number of $2$-color partitions of $n$ where one of
the colors appears only in parts that are multiples of $k$. We will prove a
conjecture of Ahmed, Baruah, and Dastidar on congruences modulo $5$ for
$p_k(n)$. Moreover, we will present some new congruences modulo $7$ for
$p_4(n)$.
