Semantic parsing is an important NLP problem, particularly for voice assistants such as Alexa and Google Assistant. State-of-the-art (SOTA) semantic parsers are seq2seq architectures based on large language models that have been pretrained on vast amounts of text. To better leverage that pretraining, recent work has explored a reformulation of semantic parsing whereby the output sequences are themselves natural language sentences, but in a controlled fragment of natural language. This approach delivers strong results, particularly for few-shot semantic parsing, which is of key importance in practice and the focus of our paper. We push this line of work forward by introducing an automated methodology that delivers very significant additional improvements by utilizing modest amounts of unannotated data, which is typically easy to obtain. Our method is based on a novel synthesis of four techniques: joint training with auxiliary unsupervised tasks; constrained decoding; self-training; and paraphrasing. We show that this method delivers new SOTA few-shot performance on the Overnight dataset, particularly in very low-resource settings, and very compelling few-shot results on a new semantic parsing dataset.