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Defect-free atom arrays are an important precursor for quantum information processing and quantum simulation. Yet, large-scale defect-free atom arrays remain challenging to realize, due to the losses encountered when rearranging stochastically loaded atoms to achieve a desired target array. Here, we demonstrate a novel parallel rearrangement algorithm that uses multiple mobile tweezers to independently sort and compress atom arrays in a way that naturally avoids atom collisions. For a high degree of parallelism, the required number of moves scales as N 0.5 with respect to the target array size N , as opposed to existing single-tweezer algorithms that scale as N or larger. We further determine the optimal degree of parallelism to be a balance between the algorithmic speedup and intermodulation effects. The defect-free probability for a 225-atom array is demonstrated to be as high as 33(1)% in a room temperature setup after multiple cycles of rearrangement. The algorithm presented here can be implemented for any target array geometry with an underlying periodic structure.