 Parabolic stabilizer of a subspace

For column vectors with the preserved space first, $A,D$ are invertible and $B$ arbitrary. Thus $P_k\cong {\mathbb{F}}_q^{k(n-k)}⋊(G{L}_k(q)\times G{L}_{n-k}(q))$. Its orbits on $k$-spaces are classified by intersection dimension with the preserved space.