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In this paper, we are interested in developing semantic parsers which understand natural language questions embedded in a conversation with a user and ground them to formal queries over definitions in a general purpose knowledge graph (KG) with very large vocabularies (covering thousands of concept names and relations, and millions of entities). To this end, we develop a dataset where user questions are annotated with Sparql parses and system answers correspond to execution results thereof. We present two different semantic parsing approaches and highlight the challenges of the task: dealing with large vocabularies, modelling conversation context, predicting queries with multiple entities, and generalising to new questions at test time. We hope our dataset will serve as useful testbed for the development of conversational semantic parsers. Our dataset and models are released at https://github.com/EdinburghNLP/SPICE.