A central challenge for realizing large-scale quantum processors is the design and realization of qubit-qubit connections: we must be able to perform efficient gates between qubits, yet prevent connections from spoiling qubit quality or prohibiting "debugging" the system. In this work, we present a microwave quantum state router that realizes all-to-all couplings among four independent and detachable quantum modules of superconducting qubits. Each module consists of a single transmon, readout mode, and communication mode coupled to the router. The router design centers on a parametrically driven, Josephson-junction based three-wave mixing element which generates photon exchange among the modules' communication modes. We first demonstrate coherent photon exchange among four communication modes, with an average full-iSWAP time of 760 ns and average inter-module gate fidelity of 0.97, limited by our modes' coherence times. We also demonstrate photon transfer and pairwise entanglement between the modules' qubits, and parallel operation of simultaneous iSWAP across the router. The gates demonstrated here can readily be extended to faster and higher-fidelity router operations, as well as scaled to support larger networks of quantum modules.