Parametricity is a property that given a parametric polymorphic function, its behaviour should be the same no matter how its type parameters are specialized.
This property is enjoyed by some (in fact, most) of the functional programming languages, but there are exceptions, such as Idris2 that supports pattern matching on types.
To some extent, parametricity is similar to a "η-rule" for polymorphic types. For instance, the type has a unique instance -- the polymorphic identity function, but this is true only if the type theory satisfies parametricity (and in a type theory that contradicts parametricity, there might be multiple distinct instances of this type).
Parametricity is necessary for proving the isomorphism between lambda-encoded inductive types and the conventional inductive types.
Wadler, Philip. 1989. ‘Theorems for Free!’ In Proceedings of the Fourth International Ponference on Functional Programming Languages and Computer Architecture - FPCA ’89, 112:347–359. New York, New York, USA: ACM Press. https://doi.org/10.1145/99370.99404. pdf
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title = {Theorems for free!},
volume = {112},
isbn = {0-89791-328-0},
doi = {10.1145/99370.99404},
booktitle = {Proceedings of the fourth {International} {Ponference} on {Functional} {Programming} languages and computer architecture - {FPCA} '89},
publisher = {ACM Press},
author = {Wadler, Philip},
year = {1989},
pages = {347--359}
}
Reynolds, John C. 1984. ‘Types, Abstraction, and Parametric Polymorphism’. In Information Processing 83: Proceedings of the IFIP 9th World Computer Congress, Paris, France, September 19-23, 1983, edited by R. E. A Mason, 513–523. Amsterdam: Elsevier Science Publishers B. V. (North-Holland). [pdf]
@inproceedings{reynolds_1983,
address = {Amsterdam},
title = {Types, {Abstraction}, and {Parametric} {Polymorphism}},
booktitle = {Information {Processing} 83: {Proceedings} of the {IFIP} 9th {World} {Computer} {Congress}, {Paris}, {France}, {September} 19-23, 1983},
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author = {Reynolds, John C.},
editor = {Mason, R. E. A},
year = {1983},
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}