This paper deals with a common due date parallel machines scheduling problem in which each job has a different tardiness penalty. The objective is to minimize the total weighted tardiness. The scheduling problem of minimizing the total weighted tardiness with a common due date on a single machine is known to be ordinary NP-hard. This is also the case for problem P m|di = d| P wiTi. A new dynamic programming algorithm is proposed to solve the 1|di = d| P wiTi scheduling problem and a fully polynomial time approximation scheme (FPTAS) is deduced from this algorithm. We show that the complexity of this FPTAS is better than existing one. These results are generalized to the parallel machines scheduling case.