We present the theoretical basis of a cavity-enhanced polarimetric scheme for the measurement of parity-nonconserving (PNC) optical rotation. We discuss the possibility of detecting PNC optical rotation in accessible transitions in metastable Xe and
Hg, and ground state I. In particular, the physics of the PNC optical rotation is presented, and we explore the lineshape effects on the expected PNC optical rotation signals. Furthermore, we present an analysis of the eigenpolarizations of the cavity-enhanced polarimeter, which is necessary for understanding the measurement procedure and the ability of employing robust background subtraction procedures using two novel signal reversals. Using recent atomic structure theoretical calculations, we present simulations of the PNC optical rotation signals for all proposed transitions, assuming a range of experimentally feasible parameters. Finally, the possibility of performing sensitive measurements of the nuclear-spin-dependent PNC effects is investigated, for the odd-neutron nuclei $^{129}$Xe and $^{199}$Hg, and the odd-proton nucleus $^{127}$I.
