Second-order sufficient optimality conditions play an important role in sensitivity analysis in nonlinear programming. Standard second-order sufficient conditions imply that the point in question is a special type of strict local minimizer, usually termed a strong minimum or strict local minimum of order two. In this paper, we derive sufficient conditions for weak sharp minima of order two, a larger class of (possibly) non-isolated minima, and show that these sufficient conditions can be used to extend some previous results on first- and second-order directional derivatives of the value function of a family of nonlinear programs with right-hand-side perturbations of the constraints.