In this paper, we propose a fast parallel algorithm, which is generalized from the parallel algorithms for solving banded linear systems, to solve sparse triangular systems. We transform the original problem into a directed graph. The solving procedure then consists of eliminating edges in this graph. The worst case time-complexity of this parallel algorithm is O(log2n) where n is the size of the coefficient matrix. When the coefficient matrix is a triangular banded matrix with bandwidth m, then the time-complexity of our algorithm is O(log(m) log(n)).