We describe the issues involved in the design and implementation of efficient parallel algorithms for solving sparse linear least squares problems on distributed-memory multiprocessors. We consider both the QR factorization method due to Golub and the method of corrected semi-normal equations due to Bjorck. The major tasks involved are sparse QR factorization, sparse triangular solution and sparse matrix-vector multiplication. The sparse QR factorization is accomplished by a parallel multifrontal scheme recently introduced. New parallel algorithms for solving the related sparse triangular systems and for performing sparse matrix-vector multiplications are proposed. The arithmetic and communication complexities of our algorithms on regular grid problems are presented. Experimental results on an Intel iPSC/860 machine are described.