Higher order parabolic boundary Harnack inequality in $C^1$ and $C^{k,\alpha}$ domains

We study the boundary behaviour of solutions to second order parabolic linear equations in moving domains. Our main result is a higher order boundary Harnack inequality in $C^1$ and $C^{k,\alpha}$ domains, providing that the quotient of two solutions vanishing on the boundary of the domain is as smooth as the boundary. As a consequence of our result, we provide a new proof of higher order regularity of the free boundary in the parabolic obstacle problem.