We present a parallel large neighborhood search framework for finding high quality primal solutions for generic Mixed Integer Programs (MIPs). The approach simultaneously solves a large number of sub-MIPs with the dual objective of reducing infeasibility and optimizing with respect to the original objective. Both goals are achieved by solving restricted versions of two auxiliary MIPs, where subsets of the variables are fixed. In contrast to prior approaches, ours does not require a starting feasible solution. We leverage parallelism to perform multiple searches simultaneously, with the objective of increasing the effectiveness of our heuristic. Comprehensive computational experiments show the efficacy of our approach as a standalone primal heuristic and when used in conjunction with an exact algorithm.