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Recent work has shown that generation from a prompted or fine-tuned language model can perform well at semantic parsing when the output is constrained to be a valid semantic representation. We introduce BenchCLAMP, a Benchmark to evaluate Constrained LAnguage Model Parsing, that includes context-free grammars for seven semantic parsing datasets and two syntactic parsing datasets with varied output representations, as well as a constrained decoding interface to generate only valid outputs covered by these grammars. We provide low, medium, and high resource splits for each dataset, allowing accurate comparison of various language models under different data regimes. Our benchmark supports evaluation of language models using prompt-based learning as well as fine-tuning. We benchmark eight language models, including two GPT-3 variants available only through an API. Our experiments show that encoder-decoder pretrained language models can achieve similar performance or surpass state-of-the-art methods for syntactic and semantic parsing when the model output is constrained to be valid.