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Equilibria computation is of great importance to many areas such as economics, control theory, and recently computer science. We focus on the computation of Nash equilibria in two-player general-sum normal form games, also called bimatrix games. One efficient method to compute these equilibria is based on enumerating the vertices of the best response polyhedrons of the two players and checking the equilibrium conditions for every pair of vertices. We design and implement a parallel algorithm for computing Nash equilibria in bimatrix games based on vertex enumeration. We analyze the performance of the proposed algorithm by performing extensive experiments on a grid computing system.