A search for short-period pulsars using the Parkes radio telescope at a frequency of 1.4 GHz has yielded four new radio pulsars. The search, which was specifically designed to detect pulsars which may have been missed by previous large-scale pulsars surveys, had a limiting sensitivity of about 1 mJy for pulsars of period greater than 10 ms and dispersion measures less than 1600/cu cm pc. It was directed at objects in the Galaxy and Magellanic Clouds likely to contain young pulsars such as supernova remnants and gamma-ray sources. Only one of the four pulsars detected, PSR 1509-58, which was previously known as an X-ray pulsar, is apparently associated with the target object, although one other pulsar-supernova remnant association is possible. The results suggest that many short-period pulsars, including some associated with supernova remnants, would be discovered by searches with improved sensitivity.