Breakthrough Listen is a 10-yr initiative to search for signatures of technologies created by extraterrestrial civilisations at radio and optical wavelengths. Here, we detail the digital data recording system deployed for Breakthrough Listen observations at the 64-m aperture CSIRO Parkes Telescope in New South Wales, Australia. The recording system currently implements two modes: a dual-polarisation, 1.125-GHz bandwidth mode for single-beam observations, and a 26-input, 308-MHz bandwidth mode for the 21-cm multibeam receiver. The system is also designed to support a 3-GHz single-beam mode for the forthcoming Parkes ultra-wideband feed. In this paper, we present details of the system architecture, provide an overview of hardware and software, and present initial performance results.