A relativistic Green's function approach to parity-violating quasielastic electron scattering is presented. The components of the hadron tensor are expressed in terms of the single particle Green's function, which is expanded in terms of the eigenfunctions of the non-Hermitian optical potential, in order to account for final state interactions without any loss of flux. Results for 12 C, 16 O, and 40 Ca are presented and discussed. The effect of the strange quark contribution to the nuclear current is investigated.