Let $GL_{n_1,\dots,n_r}$ be a parabolic subgroup of the general linear group $GL_n$ over the prime field $\mathbb F_p$ of $p$ elements. A complete set of distinct irreducible modules for $\mathbb F_p[GL_{n_1,\dots,n_r} ]$ was explicitly constructed in [7]. In this paper, we use this construction to determine the contragredient dual module of each $\mathbb F_p[GL_{n_1,\dots,n_r} ]$-irreducible module and prove that its dimension can be computed via the dimensions of some $\mathbb F_p[GL_{n_i} ]$-irreducible modules.