We consider variations and generalizations of the initial Dirichlet problem for linear second order divergence form equations of parabolic type, with vanishing initial values and non-continuous lateral data, in the setting of Lipschitz cylinders. More precisely, lateral data in adequations of the Lebesgue classes $L^p$, and a family of Sobolev-type classes are considered. We also establish some basic connections between estimates related to solvability of each of these problems. This generalizes some of the well-known works for Laplace's equation, heat equation and some linear elliptic-type equations of second order.