  We deal with the regularity problem for linear, second order parabolic
equations and systems in divergence form with measurable data over non-smooth
domains, related to variational problems arising in the modeling of composite
materials and in the mechanics of membranes and films of simple non-homogeneous
materials which form a linear laminated medium. Assuming partial BMO smallness
of the coefficients and Reifenberg flatness of the boundary of the underlying
domain, we develop a Calder\'{o}n--Zygmund type theory for such parabolic
operators in the settings of the weighted Lebesgue spaces. As consequence of
the main result, we get regularity in parabolic Morrey scales for the spatial
gradient of the weak solutions to the problems considered.
