We consider the problem of computing all Nash equilibria in bimatrix games (i.e., nonzero-sum two-player noncooperative games). Computing all Nash equilibria for large bimatrix games using single-processor computers is not feasible due to the exponential amount of time required by the existing algorithms. To be able to solve large games we need to rely on parallel computing. We present the design of two parallel algorithms for computing all Nash equilibria in bimatrix games. The first parallel algorithm computes all equilibria by searching all possible supports of mixed strategies. The second parallel algorithm computes all equilibria by enumerating the vertices of the best response polyhedrons of the two players. Both algorithms were implemented using MPICH, a portable implementation of the Message-Passing Interface standard. We present and analyze the performance obtained by executing the two parallel algorithms on a grid computing system.