DNA pattern matching is widely applied in many bioinformatics applications. The increasing volume of the DNA data exacerbates the runtime and power consumption to discover DNA patterns. In this paper, we propose a hardware-software co-design, called GenieHD, which efficiently parallelizes the DNA pattern matching task. We exploit brain-inspired hyperdimensional (HD) computing which mimics pattern-based computations in human memory. We transform inherent sequential processes of the DNA pattern matching to highly-parallelizable computation tasks using HD computing. The proposed technique first encodes the whole genome sequence and target DNA pattern to high-dimensional vectors. Once encoded, a light-weight operation on the high-dimensional vectors can identify if the target pattern exists in the whole sequence. We also design an accelerator architecture which effectively parallelizes the HD-based DNA pattern matching while significantly reducing the number of memory accesses. The architecture can be implemented on various parallel computing platforms to meet target system requirements, e.g., FPGA for low-power devices and ASIC for high-performance systems. We evaluate GenieHD on practical large-size DNA datasets such as human and Escherichia Coli genomes. Our evaluation shows that GenieHD significantly accelerates the DNA matching procedure, e.g., 44.4× speedup and 54.1× higher energy efficiency as compared to a state-of-the-art FPGA-based design.