  The recent experimental determination of the parity violating asymmetry
$A_{\rm pv}$ in ${}^{48}$Ca and ${}^{208}$Pb at Jefferson Lab is important for
our understanding on how neutrons and protons arrange themselves inside the
atomic nucleus. To better understand the impact of these measurements, we
present a rigorous theoretical investigation of $A_{\rm pv}$ in ${}^{48}$Ca and
${}^{208}$Pb and assess the associated uncertainties. We complement our study
by inspecting the static electric dipole polarizability in these nuclei. The
analysis is carried out within nuclear energy density functional theory with
quantified input. We conclude that the simultaneous accurate description of
$A_{\rm pv}$ in ${}^{48}$Ca and ${}^{208}$Pb cannot be achieved by our models
that accommodate a pool of global nuclear properties, such as masses and charge
radii, throughout the nuclear chart, and describe -- within one standard
deviation -- the experimental dipole polarizabilities $\alpha_{\rm D}$ in these
nuclei.
