  In this paper we derive $W^{1,\infty}$ and piecewise $C^{1,\alpha}$ estimates
for solutions, and their $t-$derivatives, of divergence form parabolic systems
with coefficients piecewise H\"older continuous in space variables $x$ and
smooth in $t$. This is an extension to parabolic systems of results of Li and
Nirenberg on elliptic systems. These estimates depend on the shape and the size
of the surfaces of discontinuity of the coefficients, but are independent of
the distance between these surfaces.
