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Parallel nonnegative least squares (NNLS) solvers are developed specifically for NNLS problems that arise when the Energy Conserving Sampling and Weighting hyper-reduction procedure is used when constructing a reduced-order model (ROM) from high-fidelity, finite element calculations. With this approach, nonzero entries in the NNLS solution define a subset of the finite element mesh where the nonlinear terms need to be evaluated when integrating the ROM, and their values are weighting factors for each element; hence, a sparse solution is sought to reduce the computational work of integrating the ROM. The NNLS solution does not have to be optimal but it must satisfy some criteria associated with the accuracy of the ROM. The goal, therefore, is to produce approximate NNLS solutions that are sparse but meet some tolerance. Two algorithms are considered and scalable codes are developed: one uses the Lawson and Hanson active set method and the other the projected quasi-Newton iterative method. The latter does not inherently produce sparse solutions but can be modified to promote this. Both codes are parallelized using ScaLAPACK and performance results are presented