Summary: The authors give new sufficient criteria for smooth hypoellipticity of linear second-order degenerate parabolic partial differential operators. These criteria allow Hörmander’s condition to fail exponentially fast on a submanifold of degeneracies in the underlying Euclidean space. The exponential decay rate near the submanifold is shown to be optimal. The hypotheses include Hörmander’s general condition for parabolic hypoellipticity as a special case.