Summary: The parametric minimax problem, which finds the parameter value minimizing the weight of a solution of a combinatorial maximization problem, is a fundamental problem in sensitivity analysis. Moreover, several problems in computational geometry can be formulated as parametric minimax problems. The parametric search paradigm gives an efficient sequential algorithm for a convex parametric minimax problem with one parameter if the original non-parametric problem has an efficient parallel algorithm. We consider the parametric minimax problem with \(d\) parameters for a constant \(d\), and solve it by using multidimensional version of the parametric search paradigm. As a new feature, we give a feasible region in the parameter space in which the parameter vector must be located. Typical results obtained as applications are: (1) Efficient solutions for some geometric problems, including theoretically efficient solutions for the minimum diameter bridging problem in \(d\)-dimensional space between convex polytopes. (2) Solutions for parametric polymatroid optimization problems, including an \(O(n\log n)\) time algorithm to compute the parameter vector minimizing \(k\)-largest linear parametric elements with \(d\) dimensions.