A parallel algorithm (PAA algorithm) is presented for the solution of diagonally dominant periodic tridiagonal linear systems. Its computation complexity is O(8n) and its communication complexity is O(1). But now the computation and communication complexity of the best parallel algorithm are O(17n) and O (log P) respectively. The PAA algorithm is implemented on the SGI Indy workstation cluster. The results show that the speed up has been improved linearly and the paralleling efficiency reaches to 90%.