A feedback law expressed by means of a Trotter product formula approximation of the dynamic programming equation, and which provides approximately optimal controls, is established for the control systems governed by a certain class of variational inequalities of parabolic type. To this purpose, two general Lie-Trotter formulas for the dynamic programming equation are proposed, and corresponding convergence results (generalizing previous results of the author) are proved. The framework also includes the control systems described by the parabolic obstacle problem as well as those governed by semilinear parabolic equations.