The effect of magnetic shear on electrostatic Bernstein waves is investigated. It is shown that shear leads to nonlocal eigensolutions and the convective damping of the Bernstein waves. The threshold for the parametric instability of a magnetosonic wave into two Bernstein waves in a multispecies plasma is increased by shear. For a magnetosonic heating experiment conducted on a tokamak comparable to the Doublet III device at General Atomic, the parametric instability may still be excited despite the plasma inhomogeneities provided the pump wave amplitude exceeds a critical value to be estimated in this paper.