In this paper the Parallel Pipelining SOR (ppSOR) method and a iterative linear equation solver that uses the ppSOR method is presented. This method does not require to color nodes of the graph which is associated with the coefficient matrix to be solved. Also in this paper the experimental results on a distributed memory parallel processor that is a NEC cenju-3 with 16 PEs are presented. As shown in this paper, the convergence property of ppSOR method is as same as the multi-color method, and the experimental results confirm it. With the property, SOR acceleration works as well as on a single processor or as the multi-color method. The experimental results also confirms this, and up to 10 times speed up on a 16 PE parallel processor was obtained compared to the block SOR method which also does not require to color nodes. And we obtain almost same or a little improved performance compared to the well known multi-color method. Then with this method we can solve a linear equation without coloring but a better performance.