  We consider pion production in parity-violating electron scattering (PVES) in
the presence of nucleon strangeness in the framework of partial wave analysis
with unitarity. Using the experimental bounds on the strange form factors
obtained in elastic PVES, we study the sensitivity of the parity-violating
asymmetry to strange nucleon form factors. For forward kinematics and electron
energies above 1 GeV, we observe that this sensitivity may reach about 20\% in
the threshold region. With parity-violating asymmetries being as large as tens
p.p.m., this study suggests that threshold pion production in PVES can be used
as a promising way to better constrain strangeness contributions. Using this
model for the neutral current pion production, we update the estimate for the
dispersive $\gamma Z$-box correction to the weak charge of the proton. In the
kinematics of the Qweak experiment, our new prediction reads Re$\,\Box_{\gamma
Z}^V(E=1.165\,{\rm GeV}) = (5.58\pm1.41)\times10^{-3}$, an improvement over the
previous uncertainty estimate of $\pm2.0\times10^{-3}$. Our new prediction in
the kinematics of the upcoming MESA/P2 experiment reads Re$\,\Box_{\gamma
Z}^V(E=0.155\,{\rm GeV}) = (1.1\pm0.2) \times 10^{-3}$.
