  Contributions from physics beyond the Standard Model, strange quarks in the
nucleon, and nuclear structure effects to the left-right asymmetry measured in
parity-violating (PV) electron scattering from $\nuc{12}{C}$ and the proton are
discussed. It is shown how lack of knowledge of the distribution of strange
quarks in the nucleon, as well as theoretical uncertainties associated with
higher-order dispersion amplitudes and nuclear isospin-mixing, enter the
extraction of new limits on the electroweak parameters $S$ and $T$ from these
PV observables. It is found that a series of elastic PV electron scattering
measurements using $^{4}$He could significantly constrain the $s$-quark
electric form factor if other theoretical issues are resolved. Such constraints
would reduce the associated form factor uncertainty in the carbon and proton
asymmetries below a level needed to permit extraction of interesting low-energy
constraints on $S$ and $T$ from these observables. For comparison, the much
smaller scale of $s$-quark contributions to the weak charge measured in atomic
PV is quantified. It is likely that only in the case of heavy muonic atoms
could nucleon strangeness enter the weak charge at an observable level.
