Associated to a presentable ∞-category C and an object X ∈ C is the tangent ∞-category T X C, consisting of parameterized spectrum objects over X. This gives rise to a cohomology theory, called Quillen cohomology, whose category of coefficients is T X C. When C consists of algebras over a nice ∞-operad in a stable ∞-category, T X C is equivalent to the ∞-category of operadic modules, by work of Basterra-Mandell, Schwede and Lurie. In this paper we develop the model-categorical counterpart of this identification and extend it to the case of algebras over an enriched operad, taking values in a model category which is not necessarily stable. This extended comparison can be used, for example, to identify the cotangent complex of enriched categories, an application we take up in a subsequent paper.