We study formal modelling of architectures applied on parametric component-based systems consisting of an unknown number of instances of each component. Architecture modelling is achieved by means of logics. We introduce an extended propositional interaction logic and investigate its first-order level which serves as a formal language for the interactions of parametric systems. Our logic effectively describes the execution order of interactions which is a main feature in several important architectures. We state the decidability of equivalence, satisfiability, and validity of first-order extended interaction logic formulas, and provide several examples of formulas describing well-known architectures.