In bioinformatics, alignment is an essential technique for finding similarities between biological sequences. Usually, the alignment is performed with the Smith-Waterman (SW) algorithm, a well-known sequence alignment technique of high-level precision based on dynamic programming. However, given the massive data volume in biological databases and their continuous exponential increase, high-speed data processing is necessary. Therefore, this work proposes a parallel hardware design for the SW algorithm with a systolic array structure to accelerate the forward and backtracking steps. For this purpose, the architecture calculates and stores the paths in the forward stage for pre-organizing the alignment, which reduces the complexity of the backtracking stage. The backtracking starts from the maximum score position in the matrix and generates the optimal SW sequence alignment path. The architecture was validated on Field-Programmable Gate Array (FPGA), and synthesis analyses have shown that the proposed design reaches up to 79.5 Giga Cell Updates per Second (GCPUS).