We prove Lp-parabolic a-priori estimates for $\partial_t u + \sum_{i,j=1}^d c_{ij}(t)\partial_{x_i x_j}^2 u = f $ on $\R^{d+1}$ when the coefficients $c_{ij}$ are locally bounded functions on $\R$. We slightly generalize the usual parabolicity assumption and show that still $L^p$-estimates hold for the second spatial derivatives of $u$. We also investigate the dependence of the constant appearing in such estimates from the parabolicity constant. We extend our estimates to parabolic equations involving non-degenerate Ornstein-Uhlenbeck type operators. Finally we present some open problems.