Combinatory Categorial Grammar (CCG) is a strongly lexicalized grammatical formalism, in which the vast majority of parsing decisions involve assigning a supertag to indicate the correct syntactic role. We propose an A* parsing model for CCG that exploits this characteristics, by modeling the probability of a tree through the supertags and resolving the remaining ambiguities by its syntactic dependencies. The key of our method is that it predicts the probabilities of supertags and dependency heads independently using a strong unigram model defined over bi-directional LSTMs. The factorization allows precomputation of probabilities for all possible trees for a sentence, which, combined with an A* algorithm, enables very efficient decoding. The proposed model achieves the state-of-the-art results on English and Japanese CCG parsing. In addition, we conduct Recognizing Textual Entailment (RTE) experiments by integrating the proposed parser within logic-based RTE systems. We demonstrate that such integration leads to improved performance in English RTE experiments.