In this paper, we consider the two-parallel machine scheduling problem with a non-availability interval. We aim to minimize the maximum lateness when every job has a positive tail. We show that the problem has a constant polynomial approximation algorithm. We present a dynamic programming algorithm and we show that the problem has an FPTAS (Fully Polynomial Time Approximation Algorithm). The proposed FPTAS has a strongly polynomial running time. Finally, we present some numerical experiments and we analyze the obtained results.