1. A parallel computer that processes data communications events in a parallel active messaging interface (‘PAMI’), the parallel computer comprising a plurality of compute nodes that execute a parallel application, the PAMI comprising data communications endpoints, each endpoint comprising a specification of data communications parameters for a thread of execution on a compute node, including specifications of a client, a context, and a task, the compute nodes and the endpoints coupled for data communications through the PAMI and through data communications resources, the compute nodes comprising computer processors operatively coupled to computer memory having disposed within it computer program instructions that, when executed by the computer processors, cause the parallel computer to function by: determining by an advance function, the advance function comprising a member method of a context, the advance function executing in a thread of execution on a compute node, that there are no actionable data communications events pending for the context, placing by the advance function the thread into a wait state, waiting for a subsequent data communications event for the context; responsive to occurrence of a subsequent data communications event for the context, awakening by the thread from the wait state; and processing by the advance function the subsequent data communications event now pending for the context.