Studying a critical value function \vi in parametric nonlinear programming, we recall conditions guaranteeing that \vi is a C1,1 function and derive second order Taylor expansion formulas including second-order terms in the form of certain generalized derivatives of D\vi. Several specializations and applications are discussed. These results are understood as supplements to the well-developed theory of first- and second-order directional differentiability of the optimal value function in parametric optimization.