  We consider a family of second-order parabolic systems in divergence form
with rapidly oscillating and time-dependent coefficients, arising in the theory
of homogenization. We obtain uniform interior $W^{1,p}$, H\"older, and
Lipschitz estimates as well as boundary $W^{1,p}$ and H\"older estimates, using
compactness methods. As a consequence, we establish uniform $W^{1,p}$ estimates
for the initial-Dirichlet problems in $C^{1}$ cylinders.
