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Hybridizable discretizations allow for the elimination of local degrees-of-freedom leading to reduced linear systems. In this paper, we determine and analyze an approach to construct parameter-robust preconditioners for these reduced systems. Using the framework of Mardal and Winther [Numer. Linear Algebra Appl., 18 (2011), pp. 1–40], we first determine a parameter-robust preconditioner for the full system. We then eliminate the local degrees-of-freedom of this preconditioner to obtain a preconditioner for the reduced system. However, not all reduced preconditioners obtained in this way are automatically robust. We therefore present conditions that must be satisfied for the reduced preconditioner to be robust. To demonstrate our approach, we determine preconditioners for the reduced systems obtained from hybridizable discretizations of the Darcy and Stokes equations. Our analysis is verified by numerical examples in two and three dimensions.