  We report the measurement of the parity-violating asymmetry for the inelastic
scattering of electrons from the proton, at $Q^2 = 0.082$ GeV$^2$ and $ W =
2.23$ GeV, above the resonance region. The result $A_{\rm Inel} = - 13.5 \pm
2.0 ({\rm stat}) \pm 3.9 ({\rm syst})$~ppm agrees with theoretical
calculations, and helps to validate the modeling of the $\gamma Z$ interference
structure functions $F_1^{\gamma Z}$ and $F_2^{\gamma Z}$ used in those
calculations, which are also used for determination of the two-boson exchange
box diagram ($\Box_{\gamma Z}$) contribution to parity-violating elastic
scattering measurements. A positive parity-violating asymmetry for inclusive
$\pi^-$ production was observed, as well as positive beam-normal single-spin
asymmetry for scattered electrons and a negative beam-normal single-spin
asymmetry for inclusive $\pi^-$ production.
