Given a square matrix, the permanent is a determinant-like function without signs. In this paper, we study inequalities involving permanents of certain submatrices. We first focus on a family of zero-one totally nonnegative matrices that arise as anti-adjacency matrices of unit interval orders. For these matrices, we prove a collection of inequalities comparing products of permanents of consecutive principal submatrices with products of permanents of parity-selected principal submatrices. We also study a related combinatorial problem involving two families of permutations: a Young subgroup and a set of parity alternating permutations. We construct a bijective map from the first family to the second family, and verify computationally for n at most 13 that each permutation is below its image in Bruhat order. We conjecture that this property holds for all n. If true, this would imply one of the main permanent inequalities for all totally nonnegative matrices in the balanced case. More broadly, we conjecture that the full family of inequalities holds for all totally nonnegative matrices.