The symmetries of the Standard Model dictate that for very low energies, where nucleon dynamics can be described in terms of a pionless effective field theory (EFT(/ π)), the leading-order parity-violating nucleon-nucleon Lagrangian contains five independent unknown low-energy constants (LECs). We find that imposing the approximate symmetry of QCD that appears when the number of colors N c becomes large reduces the number of independent LECs to two at leading order in the combined EFT(/ π) and large-N c expansions. We also find a relation between the two isoscalar LECs in the large-N c limit. This has important implications for the number of experiments and/or lattice calculations necessary to confirm this description of physics. In particular, we find that a future measurement of the PV asymmetry in γd → np together with the existing result for PV pp scattering would constrain all leading-order (in the combined expansion) LECs. This is considerably improved from the previous understanding of the system.