  In this paper we study the existence, the optimal regularity of solutions,
and the regularity of the free boundary near the so-called \emph{regular
points} in a thin obstacle problem that arises as the local extension of the
obstacle problem for the fractional heat operator $(\partial_t - \Delta_x)^s$
for $s \in (0,1)$. Our regularity estimates are completely local in nature.
This aspect is of crucial importance in our forthcoming work on the blowup
analysis of the free boundary, including the study of the singular set. Our
approach is based on first establishing the boundedness of the time-derivative
of the solution. This allows reduction to an elliptic problem at every fixed
time level. Using several results from the elliptic theory, including the
epiperimetric inequality, we establish the optimal regularity of solutions as
well as $H^{1+\gamma,\frac{1+\gamma}{2}}$ regularity of the free boundary near
such regular points.
