This paper aims is to give an algorithm for the slope stability of toric vector bundles (or equivariant reflexive sheaves). We also provide a visualisation of stability in terms of their parliaments of polytopes, a generalization of Newton polytopes for toric vector bundles by Di Rocco, Jabbusch, and Smith. We finally define subparliaments of polytopes and identify them with parliaments of equivariant subbundles.