We propose an efficient parallel Gauss-Seidel (GS) iterative algorithm for solving large systems of linear equations on a 3-dimensional torus network-on-chip (NoC) architecture. The proposed parallel algorithm is O(Nn2/k3) time complexity for solving a system with matrix of order n on a k×k×k 3D torus NoC architecture with N iterations assuming n and N are large compared to k. We show that under these conditions the proposed parallel GS algorithm has near optimal speedup.