Parity-violating (PV) interactions among quarks in the nucleon induce a PV γNN coupling, or anapole moment (AM). We compute electroweak gauge-independent contributions to the AM through O(1/Λχ2) in chiral perturbation theory. We estimate short-distance PV effects using resonance saturation. The AM contributions to PV electron-proton scattering slightly enhance the axial vector radiative corrections, RAp, over the scale implied by the standard model when weak quark-quark interactions are neglected. We estimate the theoretical uncertainty associated with the AM contributions to RAp to be large, and discuss the implications for the interpretation PV of ep scattering.