  We perform a series of two-dimensional 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,
the rotations law, the degree of differential rotation, and the profiles of the
magnetic fields to construct the initial conditions. By so doing, we estimate
how the rotation-induced anisotropic neutrino heating are affected by the
strong magnetic fields through parity-violating effects and first investigate
how the north-south asymmetry of the neutrino heating in a strongly magnetized
supernova core could be. 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 the neutrino leakage
scheme. With these computations, we find that the parity-violating corrections
reduce $ \lesssim 0.5 %$ of the neutrino heating rate than that without the
magnetic fields in the vicinity of the north pole of a star, on the other hand,
enhance about $ \lesssim 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 time.
