Quantum networks play an important role in the implementation of quantum computing, communication and metrology. Circuit quantum electrodynamics (QED), consisting of superconducting artificial atoms coupled to on-chip resonators, provides a prime candidate to implement these networks due to their controllability and scalability. Furthermore, recent advances have also pushed the technology to the ultrastrong coupling (USC) regime of light-matter interaction, where the qubit-cavity coupling strength reaches a considerable fraction of the cavity frequency. Here, we propose the implementation of a scalable quantum random-access memory (QRAM) architecture based on a circuit QED network, whose edges operate in the USC regime. In particular, we study the storage and retrieval of quantum information in a parity-protected quantum memory and propose quantum interconnects in experimentally feasible schemes. Our proposal may pave the way for novel quantum memory applications ranging from entangled-state cryptography, teleportation, purification, fault-tolerant quantum computation, to quantum simulations.