  The parity violating asymmetry $A_{PV}$ in $^{208}$Pb, recently measured by
the PREX-2 collaboration, is studied using modern relativistic (covariant) and
non-relativistic energy density functionals. We first assess the theoretical
uncertainty on $A_{PV}$ which is intrinsic to the adopted approach. To this
end, we use quantified functionals that are able to accommodate our previous
knowledge on nuclear observables such as binding energies, charge radii, and
the dipole polarizability $\alpha_D$ of $^{208}$Pb. We then add the quantified
value of $A_{PV}$ together with $\alpha_D$ to our calibration dataset to
optimize new functionals. Based on these results, we predict a neutron skin
thickness in $^{208}$Pb $r_\mathrm{skin} =0.19\pm 0.02$\,fm and the
symmetry-energy slope $L=54\pm 8$\,MeV. These values are consistent with other
estimates based on astrophysical data and are significantly lower than those
recently reported using a particular set of relativistic energy density
functionals. We also make a prediction for the $A_{PV}$ value in $^{48}$Ca that
will be soon available from the CREX measurement.
