As systems for quantum computing keep growing in size and number of qubits, challenges in scaling the control capabilities are becoming increasingly relevant. Efficient schemes to simultaneouslymediate coherent interactions between multiple quantum systems and to reduce decoherence errors can minimize the control overhead in next-generation quantum processors. Here, we present a superconducting qubit architecture based on tunable parametric interactions to perform two-qubit gates, reset, leakage recovery and to read out the qubits. In this architecture, parametrically driven multi-element couplers selectively couple qubits to resonators and neighbouring qubits, according to the frequency of the drive. We consider a system with two qubits and one readout resonator interacting via a single coupling circuit and experimentally demonstrate a controlled-Z gate with a fidelity of 98.30±0.23%, a reset operation that unconditionally prepares the qubit ground state with a fidelity of 99.80±0.02% and a leakage recovery operation with a 98.5±0.3% success probability. Furthermore, we implement a parametric readout with a single-shot assignment fidelity of 88.0±0.4%. These operations are all realized using a single tunable coupler, demonstrating the experimental feasibility of the proposed architecture and its potential for reducing the system complexity in scalable quantum processors.