The rank one updating formula and the parallel eliminating algorithm for linear recurrence systems ane combined fo form a MTMD parallel algorithm fo solve tridiagonal Toeplitz Cinear equations. This algorithm has a simple strictire and requires onlty a few storages as well as in-terprocessor communications. There is no special demand on the number of processors in the parallel system. The speedup can come dose to p12 comparing with the LU decomposition method (p is the number of processors). It is worth pointing out that it is proposed algcmthn that considels and organizes the parallel arithmetic in the first order linear ricurrence systems with constant coefficients.