An incompressible flow solver based on unstructured grids is implemented on a parallel distributed memory computer architecture. An important phase in the flow solver is the solution of the elliptic equations for the velocities and pressure. This elliptic solver is parallelized and incorporated into both the explicit and implicit versions of the incompressible flow solver. Performance and scalability studies are carried out on both Intel iPSC 860 and the Intel Delta prototype, and these studies show that the code is scalable. A parallelizable load balancing algorithm is developed to be used in conjunction with the incompressible flow solver. Steady and unsteady flows over a tri-element airfoil and NACA0012 airfoil are computed using the parallel incompressible flow solver.