This paper studies the solution of large-scale sparse linear least squares problems on distributed-memory multiprocessors. The method of corrected semi-normal equations is considered. New block-oriented parallel algorithms are developed for solving the related sparse triangular systems. The arithmetic and communication complexities of the new algorithms applied to regular grid problems are analyzed. The proposed parallel sparse triangular solution algorithms together with a block-oriented parallel sparse QR factorization algorithm result in a highly efficient block-oriented approach to the parallel solution of sparse linear least squares problems on distributed-memory multiprocessors. Performance of the block-oriented approach is demonstrated empirically through an implementation on an IBM Scalable POWER parallel system SP2. The largest problem solved has over two million rows and more than a quarter million columns. The execution speed for the numerical factorization of this problem achieves over 3.7 gigaflops per second on an IBM SP2 machine with 128 processors.