  We study the regularity of the free boundary in the parabolic obstacle
problem for the fractional Laplacian $(-\Delta)^s$ (and more general
integro-differential operators) in the regime $s>\frac{1}{2}$. We prove that
once the free boundary is $C^1$ it is actually $C^{2,\alpha}$.
  To do so, we establish a boundary Harnack inequality in $C^1$ and
$C^{1,\alpha}$ (moving) domains, providing that the quotient of two solutions
of the linear equation, that vanish on the boundary, is as smooth as the
boundary.
  As a consequence of our results we also establish for the first time optimal
regularity of such solutions to nonlocal parabolic equations in moving domains.
