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A parallel weighted bit-flipping (PWBF) decoding algorithm for low-density parity-check (LDPC) codes is proposed. Compared to the best known serial weighted bit-flipping decoding, the PWBF decoding converges significantly faster but with little performance penalty. For decoding of finite-geometry LDPC codes, we demonstrate through examples that the proposed PWBF decoding converges in about 5 iterations with performance very close to that of the standard belief-propagation decoding.