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Parallelization of an Implicit discontinuous Galerkin method based on Taylor series basis for the compressible flows on unstructured meshes is developed for distributed memory architectures, specifically for cost effective compute clusters. The system of linear equations arising from the implicit time integration is solved using three choices of linear solvers: SGS(k) (Symmetric Gauss-Seidel with k iterations), LU-SGS (Lower-Upper Symmetric Gauss-Seidel), and a well known Krylov subspace iterative solver GMRES (Generalized Minimum Residual) preconditioned with LUSGS. The comparative study of the parallel performance of the flow solver based on the different linear solvers is tested on a number of parallel platforms; ranging from compute clusters to multicore machines. The parallelization is based on computational domain partitioning using the well-known mesh partitioning software, METIS, and SPMD (Single Program Multiple Data) message-passing programming paradigm using MPI (Message Passing Interface) library, which is a de facto industry standard for portable programming.