The $L^p$ Regularity problem is a Dirichlet problem with data in the homogeneous Sobolev space $\dot W^{1,p}$, or in the parabolic Sobolev space $\dot L^{p}_{1,1/2}$ for parabolic operators. The $L^p$ Regularity problem is solvable if the $L^p$ norm of the non-tangential maximal function of the derivatives of solutions can be controlled. Very recently, Dindos, Pipher and I have obtained sharp solvability of the $L^p$ Regularity problem for a class of parabolic operators in divergence form with non-smooth, time-dependent coefficients that satisfy a natural and well-studied minimal smoothness assumption on cylindrical Lipschitz domains. Our method is inspired by the recent progress on the problem in the elliptic case, but requires many new ideas and techniques due to the appearance of the half derivative in time, which is a non-local operator.