1. A method of background collective operation management in a parallel computer, the parallel computer including one or more compute nodes operatively coupled for data communications over one or more data communications networks, the method comprising: determining, by a management availability module, whether a compute node in the parallel computer is available to perform a background collective operation management task; responsive to determining that the compute node is available to perform the background collective operation management task, determining, by the management availability module, whether the compute node has access to sufficient resources to perform the background collective operation management task; and responsive to determining that the compute node has access to sufficient resources to perform the background collective operation management task, initiating, by the management availability module, execution of the background collective operation management task; and wherein determining whether the compute node has access to sufficient resources to perform the background collective operation management task further comprises: identifying a predetermined threshold that specifies the amount of resources needed to perform the background collective operation management task; and determining whether the compute node has access to resources that satisfy the predetermined threshold.