We propose a new reduced basis element-cum-component mode synthesis approach for parametrized elliptic coercive partial differential equations. In the Offline stage we construct a Library of interoperable parametrized reference components relevant to some family of problems; in the Online stage we instantiate and connect reference components (at ports) to rapidly form and query parametric systems. The method is based on static condensation at the interdomain level, a conforming eigenfunction “port” representation at the interface level, and finally Reduced Basis (RB) approximation of Finite Element (FE) bubble functions at the intradomain level. We show under suitable hypotheses that the RB Schur complement is close to the FE Schur complement: we can thus demonstrate the stability of the discrete equations; furthermore, we can develop inexpensive and rigorous (system-level) a posteriori error bounds. We present numerical results for model many-parameter heat transfer and elasticity problems with particular emphasis on the Online stage; we discuss flexibility, accuracy, computational performance, and also the effectivity of the a posteriori error bounds.