  The paper investigates stability properties of solutions of optimal control
problems for semilinear parabolic partial differential equations. H\"older or
Lipschitz dependence of the optimal solution on perturbations are obtained for
problems in which the equation and the objective functional are affine with
respect to the control. The perturbations may appear in both the equation and
in the objective functional and may non-linearly depend on the state and
control variables. The main results are based on an extension of recently
introduced assumptions on the joint growth of the first and second variation of
the objective functional. The stability of the optimal solution is obtained as
a consequence of a more general result obtained in the paper -- the proved
metric subregularity of the mapping associated with the system of first-order
necessary optimality conditions. This property also enables error estimates for
approximation methods. Lipschitz estimate for the dependence of the optimal
control on the Tikhonov regularization parameter is obtained as a by-product.
