Hardware-accelerated cloud computing systems based on FPGA chips (FPGA cloud) or ASIC chips (ASIC cloud) have emerged as a new technology trend for power-efficient acceleration of various software applications. However, the operating systems and hypervisors currently used in cloud computing will lead to power, performance, and scalability problems in an exascale cloud computing environment. Consequently, the present study proposes a parallel hardware hypervisor system that is implemented entirely in special-purpose hardware, and that virtualizes application-specific multi-chip supercomputers, to enable virtual supercomputers to share available FPGA and ASIC resources in a cloud system. In addition to the virtualization of multi-chip supercomputers, the system’s other unique features include simultaneous migration of multiple communicating hardware tasks, and on-demand increase or decrease of hardware resources allocated to a virtual supercomputer. Partitioning the flat hardware design of the proposed hypervisor system into multiple partitions and applying the chip unioning technique to its partitions, the present study introduces a cloud building block chip that can be used to create FPGA or ASIC clouds as well. Single-chip and multi-chip verification studies have been done to verify the functional correctness of the hypervisor system, which consumes only a fraction of (10%) hardware resources.