You said that the only port on the printer was a parallel? I'm not sure if they make them for parallel, but I I have a strong feeling you could get a print server and plug that into the router you mentioned. You could get a cheap PC and use it as an always-on printserver, or you could buy a network-ready printer with an ethernet port on it and a built in printserver. Not sure which would work the best for you, maybe just attach the printer to one of the computers, and leave it on 24/7 and just print to it. Only problem with that is that you have a computer running 24/7.
Just trying to offer some ideas.