  We deal with linear parabolic (in sense of Petrovskii) systems of order 2b
with discontinuous principal coefficients. A'priori estimates in Sobolev and
Sobolev--Morrey spaces are proved for the strong solutions by means of
potential analysis and boundedness of certain singular integral operators with
kernels of mixed homogeneity. As a byproduct, precise characterization of the
Morrey, BMO and H\"older regularity is given for the solutions and their
derivatives up to order 2b-1. Some counterexamples of recent publications
announcing estimates similar to the obtained here are also given.
