  We consider parabolic systems in divergence form with piecewise
$C^{(s+\delta)/2,s+\delta}$ coefficients and data in a bounded domain
consisting of a finite number of cylindrical subdomains with interfacial
boundaries in $C^{s+1+\mu}$, where $s\in\mathbb N$, $\delta\in (1/2,1)$, and
$\mu\in (0,1]$. We establish piecewise $C^{(s+1+\mu')/2,s+1+\mu'}$ estimates
for weak solutions to such parabolic systems, where
$\mu'=\min\big\{1/2,\mu\big\}$, and the estimates are independent of the
distance between the interfaces. In the elliptic setting, our results answer an
open problem (c) in Li and Vogelius (Arch. Rational Mech. Anal. 153 (2000),
91--151).
