Radar is often affected by various clutter backgrounds in complex environments, so clutter suppression has important practical significance for radar target detection. The clutter suppression process conforms to the blind source separation (BSS) model. The principal skewness analysis (PSA) algorithm is a BSS algorithm with third-order statistics as the objective function, and its running speed is faster than the conventional BSS algorithm. Still, the PSA algorithm has the problem of error accumulation. This paper improves the PSA algorithm and proposes a parallel PSA (PPSA) algorithm. PPSA can estimate the directions corresponding to each independent component simultaneously and avoid the problem of error accumulation. PPSA uses parallel instead of serial computing, significantly improving the running speed. In this paper, the PPSA algorithm is applied to radar target detection. The simulation data and real data experiments verify the effectiveness and superiority of the PPSA algorithm in suppressing clutter.