  We consider divergence form, second-order strongly parabolic systems in a
cylindrical domain with a finite number of subdomains under the assumption that
the interfacial boundaries are $C^{1,\text{Dini}}$ and $C^{\gamma_{0}}$ in the
spatial variables and the time variable, respectively. Gradient estimates and
piecewise $C^{1/2,1}$-regularity are established when the leading coefficients
and data are assumed to be of piecewise Dini mean oscillation or piecewise
H\"{o}lder continuous. Our results improve the previous results in
\cite{ll,fknn} to a large extent. We also prove a global weak type-$(1,1)$
estimate with respect to $A_{1}$ Muckenhoupt weights for the parabolic systems
with leading coefficients which satisfy a stronger assumption. As a byproduct,
we give a proof of optimal regularity of weak solutions to parabolic
transmission problems with $C^{1,\mu}$ or $C^{1,\text{Dini}}$ interfaces. This
gives an extension of a recent result in \cite{css} to parabolic systems.
