For each $\delta > 0$ there is a parabolic operator in the half-plane $R_x \times R^+_t$ whose parabolic measure is supported by a boundary set of dimension $< \delta$.
Robert Kaufman. Jang-Mei Wu. "An Example on Highly Singular Parabolic Measure." Ann. Probab. 16 (4) 1821 - 1831, October, 1988. https://doi.org/10.1214/aop/1176991599