  We obtain the regularity of solutions in Sobolev spaces for the mixed
Dirichlet-conormal problem for parabolic operators in cylindrical domains with
time-dependent separations, which is the first of its kind. Assuming the
boundary of the domain to be Reifenberg-flat and the separation to be locally
sufficiently close to a Lipschitz function of $m$ variables, where
$m=0,\ldots,d-2$, with respect to the Hausdorff distance, we prove the unique
solvability for $p\in (2(m+2/(m+3),2(m+2)/(m+1)))$. In the case when $m=0$, the
range $p\in(4/3,4)$ is optimal in view of the known results for Laplace
equations.
