The amount of mass in small, dark matter clumps within galaxies (substructure) is an important test of cold dark matter. One approach to measuring the substructure mass fraction is to analyze the fluxes of images that have been strongly lensed by a galaxy. Flux ratios between images that are anomalous with respect to smooth (no substructure) models have previously suggested that there is a greater amount of substructure than found in dark matter simulations. One measure of anomalous flux ratios is parity dependence - that the fluxes of different images of a source are perturbed differently. In this paper, we discuss parity dependence as a probe of dark matter substructure. We find that reproducing the observed parity dependence requires a significant alignment between concentrated dark matter clumps and images. The results may imply a greater fraction of mass in substructures than suggested by some dark matter simulations and that the observed parity dependence is unlikely to be reproduced by luminous satellites of lens galaxies.