Partial regularity of solutions to a class of 2m-order quasilinear parabolic systems and full interior regularity for 2m-order linear parabolic systems with non smooth in time principal matrices is proved in the paper. The coefficients are assumed to be bounded and measurable in the time variable and VMO-smooth in the space variables uniformly with respect to time. To prove the result, we apply the (A(t), m)-caloric approximation method, m >= 1. It is both an extension of the A(t)-caloric approximation applied by the authors earlier to study regularity problem for systems of the second order with non-smooth coefficients and an extension of the A-polycaloric lemma proved by V. Bogelein in [6] to systems of 2m-order.