This paper studies the second-order properties of a class of inequality-constrained bilevel programming problems. First-order optimality conditions for the existence of solutions to bilevel optimization problems are derived using the first-order directional derivative of the optimal solution function of the lower-level problem in the seminal paper by Dempe (1992). In this work, we prove that the optimal solution function of the lower-level problem is a parabolic second-order directionally differentiable function under certain assumptions. The associated second-order necessary and sufficient optimality conditions for the bilevel problem are derived. In this process, the lower-level problem may admit multiple KKT multiplier vectors.