We report a new measurement of the parity-violating asymmetry in elastic electron scattering from the proton at backward scattering angles. This asymmetry is sensitive to the strange magnetic form factor of the proton as well as electroweak axial radiative corrections. The new measurement of A = -4.92 +/- 0.61 +/- 0.73 ppm provides a significant constraint on these quantities. The implications for the strange magnetic form factor are discussed in the context of theoretical estimates for the axial corrections.