We consider a weak solution to the non-linear, parabolic systems of the form where the vector field A satisfies a Dini-type continuity condition with respect to the variables (x, t, u), and we prove a partial regularity result for such a solution. Moreover, we give an estimate of the size of the singular set of a solution in terms of a generalized parabolic Hausdorff measure associated to an appropriate modulus of continuity naturally associated to the coefficients of the system. Date: October 6, 2012.