Solving sparse systems of linear equations is of great importance when dealing with computational fluid dynamics (CFD). The Conjugate gradient (CG) method is one of the most famous iterative methods for solving such systems. Due to large problem sizes in real-world applications, we need to parallelize the CG method to scale for large systems of linear equations and run on large compute clusters. To accomplish this task, we need software libraries that enable us to express the parallelization of the algorithm at a shared and distributed memory level. The objective of this master’s thesis is to present a comparative analysis between the industry-standard tools MPI + OpenMP and the task-based parallel framework HPX in terms of inter-node and intra-node levels of task parallelization. The comparison offers insights on how to express the parallelization of the CG method with both MPI+OpenMP and HPX. Next, simulating the classical benchmark problem of lid-driven cavity, we provide performance evaluations of the two software methodologies for both shared and distributed memory systems using the implemented parallel CG method. Finally, we provide an analysis of the results and present the advantages and disadvantages of both methods.