  We prove $L^p$-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. Finally we extend our estimates to parabolic equations
involving non-degenerate Ornstein-Uhlenbeck type operators.
