An efficient assignment of tasks to the processors is imperative for achieving a fast job turnaround time in a parallel or distributed environment. The assignment problem is well known to be NP-complete, except in a few special cases. Thus heuristics are used to obtain suboptimal solutions in reasonable amount of time. While a plethora of such heuristics have been documented in the literature, in this paper we aim to develop techniques for finding optimal solutions under the most relaxed assumptions. We propose a best-first search based parallel algorithm that generates optimal solution for assigning an arbitrary task graph to an arbitrary network of homogeneous or heterogeneous processors. The parallel algorithm running on the Intel Paragon gives optimal assignments for problems of medium to large sizes. We believe our algorithms to be novel in solving an indispensable problem in parallel and distributed computing.