Large amounts of data are essential for training statistical machine translation systems. In this paper we show how training data can be expanded by paraphrasing one side of a parallel corpus. The new data is made by parsing then generating using an open-source, precise HPSG-based grammar. This gives sentences with the same meaning, but with minor variations in lexical choice and word order. In experiments paraphrasing the English in the Tanaka Corpus, a freely-available Japanese-English parallel corpus, we show consistent, statistically-significant gains on training data sets ranging from 10,000 to 147,000 sentence pairs in size as evaluated by the BLEU and METEOR automatic evaluation metrics.