We present a detailed study of the effect of the double core polarization (the polarization of the core electrons due to the simultaneous action of the electric dipole and parity-violating weak fields) for amplitudes of the ss and sd parity non-conserving transitions in Rb, Cs, Ba + , La 2+ , Tl, Fr, Ra + , Ac 2+ and Th 3+ as well as electron EDM enhancement factors for the ground states of the above neutral atoms and Au. This effect is quite large and has the potential to resolve some disagreement between calculations in the literature. It also has significant consequences for the use of experimental data in the accuracy analysis.