Three Adjoining Grammar (TAG) is a powerful grammatical formalism for large-scale natural language processing. However, the computational complexity of parsing algorithms for TAG is high. We introduce a new parallel TAG parsing algorithm for MIMD hypercube multicomputers, using large granularity grammar partitioning, asynchronous communication, and distributed termination detection. We describe our implementation on the nCUBE/2 parallel computer, and provide experimental results on both random and English grammars. Our algorithm delivers the best performance of any TAG parsing algorithm to date, yielding an almost two order-of-magnitude speedup and good efficiency on up to 256 processors.