In the solution of (parametrized) PDEs and non-linear eigenvalue problems from physics and engineering applications, often large sets of functions have to be approximated, containing singularities in either the spatial or frequency domain. This typically results in a prohibitively large computational and memory cost. Both can be reduced by switching to set-valued approximation. Here, we present a fast and parallel set-valued rational approximation algorithm for (parametrized) PDEs. Our method, called PQR-AAA, is a set-valued variant of the Adaptive Antoulas Anderson (AAA) algorithm, accelerated by using local approximate orthogonal bases obtained from a truncated QR decomposition. We demonstrate its connection to interpolative decompositions and present various theoretical insights. We focus in particular on gluing together separate approximations built in parallel, while maintaining prescribed accuracy. We demonstrate both theoretically and numerically this method's accuracy and effectiveness. Various reference problems and real-world 3D applications are presented. This talk is based on joint work with Daan Huybrechs and Karl Meerbergen.