  A class of parametric optimal control problems governed by semilinear
parabolic equations with mixed pointwise constraints is investigated. The
perturbations appear in the objective functional, the state equation and in
mixed pointwise constraints. By analyzing regularity and establishing stability
condition of Lagrange multipliers we prove that, if the strictly second-order
sufficient condition for the unperturbed problem is valid, then the solutions
of the problems as well as the associated Lagrange multipliers are locally
Lipschitz continuous functions of parameters.
