BIP is a component-based framework for system design that has important industrial applications. BIP is built on three pillars: behavior, interaction, and priority. In this paper, we introduce first-order interaction logic (FOIL) that extends BIP to systems parameterized in the number of components. We show that FOIL captures classical parameterized architectures such as token-passing rings, cliques of identical components communicating with rendezvous or broadcast, and client-server systems. Although the BIP framework includes efficient verification tools for statically-defined systems, none are available for parameterized systems with an unbounded number of components. The parameterized model checking literature contains a wealth of techniques for systems of classical architectures. However, application of these results requires a deep understanding of parameterized model checking techniques and their underlying mathematical models. To overcome these difficulties, we introduce a framework that automatically identifies parameterized model checking techniques applicable to a BIP design. To our knowledge, it is the first framework that allows one to apply prominent parameterized model checking results in a systematic way.