As I mentioned, I set up a Workgroup. I don't know if this makes any difference, because I certainly 'Removed Devices' - the existing HP Jet 4s - for both remotes, and I've just looked in Devices and Printers, and it's showing HP Laserjet 4 on Rivets-PC1 only.
PC1 is the one with the parallel port.
In a fit of . . . determination, on the laptop I removed everything before asking it to 'Add a Network Wireless or Bluetooth Printer.' In each case, it took a moment to search then showed PC1 which has become a kind of printer hub. Once it did that, it worked straight away.
(While I was making the error of using the same 'name' for the computers, it would not offer the other computer as a contender to highlight.)
Prior to this, my wife has often set up her computers to use the printers in my den. I'm almost certain we did not set up a Workgroup, so no doubt we would have had to have the printer installed on the remote units to do it by another method.