  We analyze the dispersion correction to elastic parity violating
electron-proton scattering due to $\gamma Z$ exchange. In particular, we
explore the theoretical uncertainties associated with modeling contributions of
hadronic intermediate states. Taking into account constraints from low- and
high-energy, parity-conserving electroproduction measurements, choosing
different models for contributions from the non-resonant processes, and
performing the corresponding flavor rotations to obtain the electroweak
amplitude, we arrive at an estimate of the uncertainty in the total
contribution to the parity-violating asymmetry. At the kinematics of the Q-Weak
experiment, we obtain a correction to the asymmetry equivalent to a shift in
the proton weak charge of $(0.0054\pm0.0020)$. This should be compared to the
value of the proton's weak charge of $\qwp=0.0713\pm0.0008$ that includes SM
contributions at tree level and one-loop radiative corrections. Therefore, we
obtain a new Standard Model prediction for the parity-violating asymmetry in
the kinematics of the Q-Weak experiment of $(0.0767\pm0.0008\pm0.0020_{\gamma
Z})$. The latter error leads to a relative uncertainty of 2.8% in the
determination of the proton's weak charge, and is dominated by the uncertainty
in the isospin structure of the inclusive cross section. We argue that future
parity-violating inelastic $ep$ asymmetry measurements at low-to-moderate $Q^2$
and $W^2$ could be exploited to reduce the uncertainty associated with the
dispersion correction. Because the corresponding shift and error bar decrease
monotonically with decreasing beam energy, a determination of the proton's weak
charge with a lower-energy experiment or measurements of "isotope ratios" in
atomic parity-violation could provide a useful cross check on any implications
for physics beyond the Standard Model derived from the Q-Weak measurement.
