  We study the pointwise regularity of solutions to parabolic equations. As a
first result, we prove that if the modulus of mean oscillation of $\Delta u
-u_t$ at the origin is Dini (in $L^p$ average), then the origin is a Lebesgue
point of continuity (still in $L^p$ average) for $D^2 u$ and $\dd_t u$. We
extend this pointwise regularity result to the parabolic obstacle problem with
Dini right hand side. In particular, we prove that the solution to the obstacle
problem has, at regular points of the free boundary, a Taylor expansion up to
order two in space and one in time (in the $L^p$ average). Moreover, we get a
quantitative estimate of the error in this Taylor expansion. Our method is
based on decay estimates obtained by contradiction, using blow-up arguments and
Liouville type theorems. As a by-product of our approach, we deduce that the
regular points of the free boundary are locally contained in a $C^1$
hypersurface for the parabolic distance $\sqrt{x^2 +|t|}$.
