With the contemporary emerging transition for the past two decades in high-performance computing (HPC) towards clusters of nodes, it is possible to obtain an efficiently scalable computational fluid dynamics (CFD) code by the hybridization of a message-passing interface (MPI) and open multi-processing (Open-MP). The parallelization of an incompressible higher-order direct numerical simulation (DNS) solver using Navier-Stokes (NS) equations utilizing a weighted essentially non-oscillatory (WENO) scheme is achieved through a hybrid parallel-model MPI and Open-MP. In order to further improve the performance, extensive analysis, optimization, and refactoring of the algorithms for fine tuning to utilize the resources efficiently is presented in this thesis. Initially, a time-dependent two-dimensional diffusion equation was parallelized using an MPI in an effort to explore the concepts. Subsequently, the procedure is extended with several advanced MPI routines for the parallelization of higher-order finite-difference schemes. The parallelized code is subsequently examined on a benchmark problem for an incompressible flow-lid-driven cavity flow. The results of the developed code were compared and validated with the existing serial model. The overall objective is to achieve the best performance of the incompressible Navier-Stokes solver by carefully implementing, profiling, and optimizing a combined MPI and Open-MP parallelization. Numerical simulation results show that this solver provides high performance and good scalability on various parallel architectures. MPI processes are limited by the number of communicators and grid size of the simulation. However, fine-grain parallelization is possible by incorporating Open-MP threads in certain segments of the code. Further, this computation-intensive CFD code is a challenging workload for HPC, and efficient tuning approaches are also presented.