That sounds like a pretty ancient computer.
The printer's obviously fine, and you say there were documents in the print queue, so it sounds like a communications problem between the printer and the parallel port.
The first thing to check is the port itself - make sure no pins are bent or missing. Then do the same at the printer end. If all is well, and the cable is fine - which it will be if it worked on another machine - there's almost certainly a driver issue. Try removing the old HP driver, and install the latest version for the printer; you can download that from the HP site.