Parity-violating (PV) nucleon-nucleon ($N\!N$) interactions at low energies can be completely characterized by five low energy constants (LECs). The field of hadronic parity violation has long been plagued by uncertainties in these five LECs. Experiments to reliably extract these LECs are difficult as they involve few nucleon systems which give PV asymmetries roughly of the size $10^{-8}$ to $10^{-7}$. Theoretically determining the low energy constants from the fundamental interactions of QCD is arguably more difficult, being hindered by the non-perturbative nature of QCD at low energies. In light of these facts, a theoretically motivated organizing principle for the relative size of the five LECs to guide future experiments would be highly desirable. Such an organizing principle has recently been provided by a large-$N_{C}$ (number of colors) analysis of QCD, which is the subject of this paper. This large-$N_{C}$ analysis serves both as a testable prediction from QCD and a possible organizing principle to assess the feasibility of future experiments.