  We establish several gradient estimates for second-order divergence type
parabolic and elliptic systems. The coefficients and data are assumed to be
H\"older or Dini continuous in the time variable and all but one spatial
variables. This type of systems arises from the problems of linearly elastic
laminates and composite materials. For the proof, we use Campanato's approach
in a novel way. Non-divergence type equations under a similar condition are
also discussed.
