A family of parameter dependent optimal control problems $(O)_{h}$ with smooth data for nonlinear ODEs is considered. The problems are subject to pointwise mixed control-state constraints. It is assumed that, for a reference value h₀ of the parameter, a solution of $(O)_{h₀}$ exists. It is shown that if (i) independence, controllability and coercivity conditions are satisfied at the reference solution, then (ii) for each h from a neighborhood of h₀, a locally unique solution to $(O)_{h}$ and the associated Lagrange multiplier exist, are Lipschitz continuous and Bouligand differentiable functions of the parameter. If, in addition, the dependence of the data on the parameter is strong, then (ii) implies (i).