  We show that the quotient of two caloric functions which vanish on a portion
of an $H^{k+ \alpha}$ regular slit is $H^{k+ \alpha}$ at the slit, for $k \geq
2$. In the case $k=1$, we show that the quotient is in $H^{1+\alpha}$ if the
slit is assumed to be space-time $C^{1, \alpha}$ regular. This can be thought
of as a parabolic analogue of a recent important result in [DSS14a], whose
ideas inspired us. As an application, we show that the free boundary near a
regular point of the parabolic thin obstacle problem studied in [DGPT] with
zero obstacle is $C^{\infty}$ regular in space and time.
