The thermodynamic properties of polyelectrolyte solutions are of long-standing interest. Theoretical complexity arises not only from the long-ranged electrostatic interaction but also from the multicomponent nature of the solution, In this work, we report grand canonical Monte Carlo simulations for the effect of added salt on the osmotic pressure of a primitive model of polyelectrolyte solutions. The polymer chains are freely jointed charged hard spheres, and counterions and co-ions are charged hard spheres. We use an ensemble that allows us to calculate directly the osmotic pressure for a solution in equilibrium with a bulk salt solution. As the bulk salt concentration is increased, the concentration of salt in the polyelectrolyte solution decreases and for semidilute solutions the salt concentration is very low. In dilute solution, the salt contribution to the osmotic pressure arises from electrostatic screening and excluded volume interactions. Semidilute solutions behave like salt-free solutions. The simulations show that both polymer molecules and small ions make a significant contribution to the osmotic pressure, thus questioning theories that ignore the polymer contribution. The latter effect results in the decrease in magnitude and a strong concentration dependence of the osmotic pressure. The simulation results are in qualitative accord with experiments on DNA. Scaling theories for the osmotic pressure, however, are not in agreement with the simulations or experiments.