We introduce a new approach to training a semantic parser that uses textual entailment judgements as supervision. These judgements are based on high-level inferences about whether the meaning of one sentence follows from another. When applied to an existing semantic parsing task, they prove to be a useful tool for revealing semantic distinctions and background knowledge not captured in the target representations. This information is used to improve the quality of the semantic representations being learned and to acquire generic knowledge for reasoning. Experiments are done on the benchmark Sportscaster corpus (Chen and Mooney, 2008), and a novel RTE-inspired inference dataset is introduced. On this new dataset our method strongly outperforms several strong baselines. Separately, we obtain state-of-the-art results on the original Sportscaster semantic parsing task.