We propose a parallel algorithm for computing a threshold incomplete LU (ILU) factorization. The main idea is to interleave a parallel fixed-point iteration that approximates an
incomplete factorization for a given sparsity pattern with a procedure that adjusts the pattern. We describe and test a strategy for identifying nonzeros to be added and nonzeros to be removed from the sparsity pattern. The resulting pattern may be different and more effective than that of existing threshold ILU algorithms. Also in contrast to other parallel threshold ILU algorithms, much of the new algorithm has fine-grained parallelism.