The design of a VME-based parallel-processing data-acquisition system for the proposed TRIUMF/UBC positron emission tomographs is described. Event records, consisting of eight bytes of data, must be acquired and digitally processed at rates of 250000 to 500000 events per second, corresponding to a data acquisition rate of up to 4 Mb/s. Real-time processing is necessary to reduce the data storage required by half to about 80-90 Mb for a typical patient run, and it also reduces the average data flow and directly reduces the time required for image reconstruction. Real-time data acquisition and processing requirements can be met by about 40 20-MHz Motorola 6820/68881 processors operating asynchronously and in parallel. The authors present the system architecture, the software design, and the hardware design that realizes this very high performance.>