We propose a novel branch‑and‑bound algorithm that constructs rigorous outer approximations of the optimal point mapping for parametric optimization problems with guaranteed feasibility and optimality tolerances. The method uses the improvement‑function reformulation to define discarding and inclusion tests on sub-boxes, constructing a rigorous outer approximation. Under the same regularity conditions that ensure exactness of this reformulation, we prove finite termination of the resulting algorithm (P-ICGO). We provide an implementation in Julia and evaluate the algorithm in two studies: first, we illustrate the approximation behavior on some visual examples; second, we compute the follower’s optimal point mapping on 99 bilevel optimization problems with up to four variables.