  We extend from operators with one derivative to operators with three
derivatives the analysis of two-body hadronic parity violation in a combined
pionless effective field theory (EFT$_{\pi\!/}$) and large-$N_c$ expansion,
where $N_c$ is the number of colors in quantum chromodynamics (QCD). In elastic
scattering, these operators contribute to $S$-$P$ and $P$-$D$ wave transitions,
with five operators and their accompanying low energy coefficients (LECs)
characterizing the $S$-$P$ transitions and six operators and LECs those in
$P$-$D$ transitions. We show that the large-$N_c$ analysis separates them into
leading order in $N_c$, next-to-leading order in $N_c$, etc. Relationships
among EFT$_{\pi\!/}$ LECs emerge in the large-$N_c$ expansion. We also discuss
the renormalization scale dependence of these LECs. Our analysis can complement
lattice QCD calculations and help prioritize future parity-violating
experiments.
