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The parity-violating (PV) asymmetry of inclusive π^- production in electron scattering from a liquid deuterium target was measured at backward angles. The measurement was conducted as a part of the G0 experiment, at a beam energy of 360 MeV. The physics process dominating pion production for these kinematics is quasi-free photoproduction off the neutron via the Δ⁰ resonance. In the context of heavy-baryon chiral perturbation theory (HBχPT), this asymmetry is related to a low energy constant d_Δ^- that characterizes the parity-violating γNΔ coupling. Zhu et al. calculated d_Δ^- in a model benchmarked by the large asymmetries seen in hyperon weakmore » radiative decays, and predicted potentially large asymmetries for this process, ranging from A_γ^- = -5.2 to +5.2 ppm. The measurement performed in this work leads to A_γ^- = -0.36 ± 1.06 ± 0.37 ± 0.03 ppm (where sources of statistical, systematic and theoretical uncertainties are included), which would disfavor enchancements considered by Zhu et al. proportional to V_ud/V_us. The measurement is part of a program of inelastic scattering measurements that were conducted by the G0 experiment, seeking to determine the N-Δ axial transition form-factors using PV electron scattering.« less