Current supervised parsers are limited by the size of their labelled training data, making improving them with unlabelled data an important goal. We show how a state-of-the-art CCG parser can be enhanced, by predicting lexical categories using unsupervised vector-space embeddings of words. The use of word embeddings enables our model to better generalize from the labelled data, and allows us to accurately assign lexical categories without depending on a POS-tagger. Our approach leads to substantial improvements in dependency parsing results over the standard supervised CCG parser when evaluated on Wall Street Journal (0.8%), Wikipedia (1.8%) and biomedical (3.4%) text. We compare the performance of two recently proposed approaches for classification using a wide variety of word embeddings. We also give a detailed error analysis demonstrating where using embeddings outperforms traditional feature sets, and showing how including POS features can decrease accuracy.