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

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