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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 can be 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. With a high degree of parallelism, our algorithm offers a reduced move complexity compared to both single-tweezer algorithms and existing multi-tweezer algorithms. We further determine the optimal degree of parallelism to be a balance between an algorithmic speedup and multi-tweezer inhomogeneity 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.