To parse a string according to a grammar means to reconstruct the production tree (or trees) that indicate how the given string can be produced from the given grammar. It is significant in this respect that one of the first publications on parsing (Greibach’s 1963 doctoral thesis [6]), was titled “Inverses of Phrase Structure Generators”, where a phrase structure generator is to be understood as a system for producing phrases from a phrase structure (actually context-free) grammar.
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Grune, D., Jacobs, C.J.H. (2008). Introduction to Parsing. In: Parsing Techniques. Monographs in Computer Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-68954-8_3
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