We present the reweighting of two sets of nuclear parton distribution functions (PDFs), nCTEQ15 and EPPS16, using a selection of experimental data on heavy-flavor meson [$D^0$, $J/\psi$, $B\to J/\psi$ and $\mathrm{\Upsilon }(1S)$] production in proton-lead collisions at the LHC which were not used in the original determination of these nuclear PDFs. The reweighted PDFs exhibit significantly smaller uncertainties thanks to these new heavy-flavor constraints. We present a comparison with another selection of data from the LHC and relativistic heavy ion collider (RHIC) which were not included in our reweighting procedure. The comparison is overall very good and serves as a validation of these reweighted nuclear PDF sets, which we dub $\mathrm{nCTEQ}{15}_{\mathrm{rwHF}}$ and $\mathrm{EPPS}{16}_{\mathrm{rwHF}}$. This indicates that the LHC and forward RHIC heavy-flavor data can be described within the standard collinear factorization framework with the same (universal) small-$x$ gluon distribution. We discuss how we believe such reweighted PDFs should be used as well as the limitations of our procedure.