  In this paper, we study $k$-parabolic arrangements, a generalization of the
$k$-equal arrangement for any finite real reflection group. When $k=2$, these
arrangements correspond to the well-studied Coxeter arrangements. We construct
a cell complex $Perm_k(W)$ that is homotopy equivalent to the complement. We
then apply discrete Morse theory to obtain a minimal cell complex for the
complement. As a result, we give combinatorial interpretations for the Betti
numbers, and show that the homology groups are torsion free. We also study a
generalization of the Independence Complex of a graph, and show that this
generalization is shellable when the graph is a forest. This result is used in
studying $Perm_k(W)$ using discrete Morse theory.
