This study aims to apply the Durbin-Willshaw elastic net using parallel algorithms in order to solve the Traveling Salesman Problem (TSP) through a Beowulf cluster architecture for High-Performance Computing. The solutions for the TSP for the different number of cities are achieved by the minimization of the internal energy and by the maximization of the entropy in the information system. In this way, approximate solutions to the TSP can be determined. This work proposes a framework to implement a parallel algorithm to the Beowulf cluster. In order to find solutions for the TSP, we worked with 5000 cities with a net of 12500 nodes up to 10000 cities with 25000 nodes.