We are interested in scheduling tasks from several selfish agents on a set of parallel identical machines. A coordination mechanism consists in giving a scheduling policy to each machine. Given these policies, each agent chooses the machines on which she assigns her tasks, and her aim is to minimize the average completion times of her tasks. The aim of the system (social cost) is to minimize the average completion time of all the tasks. We focus on coordination mechanisms inducing Nash equilibria, and on the performance of such mechanisms. When the machines do not know the owners of the tasks, the classical coordination mecanisms used for single-task agents do not work anymore and we give necessary conditions to obtain coordination mechanisms that induce Nash equilibria. When each machine is able to know the owner of each task it has to schedule, we give coordination mechanisms which always induce Nash equilibria.