We describe the development of a data-level, massively parallel software system for the solution of multicommodity network flow problems. Using a smooth linear-quadratic penalty (LQP) algorithm we transform the multicommodity network flow problem into a sequence of independent min-cost network flow subproblems. The solution of these problems is coordinated via a simple, dense, nonlinear master program to obtain a solution that is feasible within some user-specified tolerance to the original multicommodity network flow problem. Particular emphasis is placed on the mapping of both the subproblem and master problem data to the processing elements of a massively parallel computer, the Connection Machine CM-2. As a result of this design we can solve large and sparse optimization problems on current SIMD massively parallel architectures. Details of the implementation are reported, together with summary computational results with a set of test problems drawn from a Military Airlift Command application.