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In this paper, we propose a distributed fault localization protocol, called parallel limited perimeter vector matching (P-LVM) protocol, for localizing multi-link failures in all-optical networks. The P-LVM protocol is based on the limited perimeter vector matching (LVM) protocol for localizing single-link failures in all-optical networks. To handle multi-link failures, it tries to separate each failure in a small perimeter area after identifying each perimeter area with its corresponding failure and then localize the failures in parallel respectively in a distributed manner. Through simulation results, we show that the P-LVM protocol can effectively localize multi-link failures that occur simultaneously or at different times in all-optical networks.