We perform a series of magnetohydrodynamic simulations of supernova cores. Since the distributions of the angular momentum and the magnetic fields of strongly magnetized stars are quite uncertain, we systematically change the combinations of the strength of the angular momentum, rotation law, degree of differential rotation, and profiles of the magnetic fields to construct the initial conditions. By so doing, we estimate how the rotation-induced anisotropic neutrino heating is affected by the strong magnetic fields through parity-violating effects and for the first time investigate how the north-south asymmetry of the neutrino heating in a strongly magnetized supernova core could be affected. As for the microphysics, we employ a realistic equation of state based on the relativistic mean field theory and take into account electron captures and the neutrino transport via a neutrino leakage scheme. With these computations, we find that the neutrino heating rates are reduced by 0.5% over those without the magnetic fields as a result of the parity-violating effects in the vicinity of the north pole of a star, while they are enhanced by about 0.5% in the vicinity of the south pole. If the global asymmetry of the neutrino heating in the both of the poles develops in the later phases, the newly born neutron star might be kicked toward the north pole in the subsequent period.
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