Endpoint-based parallel data processing with non-blocking collective instructions in a parallel active messaging interface (‘PAMI’) of a parallel computer, the PAMI composed of data communications endpoints, each endpoint including 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 coupled for data communications through the PAMI, including establishing by the parallel application a data communications geometry, the geometry specifying a set of endpoints that are used in collective operations of the PAMI, including associating with the geometry a list of collective algorithms valid for use with the endpoints of the geometry; registering in each endpoint in the geometry a dispatch callback function for a collective operation; and executing without blocking, through a single one of the endpoints in the geometry, an instruction for the collective operation.