We have conducted a new search for radio pulsars in compact binary systems in the Parkes multi-beam pulsar survey (PMPS) data, employing novel methods to remove the Doppler modulation from binary motion. This has yielded unparalleled sensitivity to pulsars in compact binaries. The required computation time of approximately 17000 CPU core years was provided by the distributed volunteer computing project Einstein@Home, which has a sustained computing power of about 1 PFlop/s. We discovered 24 new pulsars in our search, of which 18 were isolated pulsars, and six were members of binary systems. Despite the wide filterbank channels and relatively slow sampling time of the PMPS data, we found pulsars with very large ratios of dispersion measure (DM) to spin period. Among those is PSR J1748-3009, the millisecond pulsar with the highest known DM (approximately 420 pc/cc). We also discovered PSR J1840-0643, which is in a binary system with an orbital period of 937 days, the fourth largest known. The new pulsar J1750-2536 likely belongs to the rare class of intermediate-mass binary pulsars. Three of the isolated pulsars show long-term nulling or intermittency in their emission, further increasing this growing family. Our discoveries demonstrate the value of distributed volunteer computing for data-driven astronomy and the importance of applying new analysis methods to extensively searched data.