A vertex signature $$\pi $$ π of a finite graph G is any mapping $$\pi \,{:}\,V(G)\rightarrow \{0,1\}$$ π:V(G)→{0,1} . An edge-coloring of G is said to be vertex-parity for the pair $$(G,\pi )$$ (G,π) if for every vertex v each color used on the edges incident to v appears in parity accordance with $$\pi $$ π , i.e. an even or odd number of times depending on whether $$\pi (v)$$ π(v) equals 0 or 1, respectively. The minimum number of colors for which $$(G,\pi )$$ (G,π) admits such an edge-coloring is denoted by $$\chi '_p(G,\pi )$$ χp′(G,π) . We characterize the existence and prove that $$\chi '_p(G,\pi )$$ χp′(G,π) is at most 6. Furthermore, we give a structural characterization of the pairs $$(G,\pi )$$ (G,π) for which $$\chi '_p(G,\pi )=5$$ χp′(G,π)=5 and $$\chi '_p(G,\pi )=6$$ χp′(G,π)=6 . In the last part of the paper, we consider a weaker version of the coloring, where it suffices that at every vertex, at least one color appears in parity accordance with $$\pi $$ π . We show that the corresponding chromatic index is at most 3 and give a complete characterization for it.