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