We establish infinite families of congruences modulo arbitrary powers of $2$ for the three restricted partition functions $M(n), T^\ast(n)$, and $P^\ast(n)$ introduced by Pushpa and Vasuki by employing elementary $q$-series techniques. These generalize some particular congruences for $M(n), T^\ast(n)$, and $P^\ast(n)$ recently found by Nath and Saikia.