Okay, call this wierd...
I found a different USB to Parallel adapter, instead of trying to directly convert the printer it installs itself as a USB IEEE1284 adapter. (So it's a USB parallel port.) This also shows up in the "ports and devices" but as a 1284 adapter. Try the printer. Bingo.
Here's the wierd part. After Windows 7 installed the drivers for this device, I then unplugged it, and plugged the printer back into the PCI parallel port. I changed the printer that I installed (On the USB001 device) to the LPT3 device, and bingo. It still prints. It continues to work even with the USB adapter unplugged.
So I strongly suspect that there was some sort of missing driver for a parallel port missing, which Windows 7 didn't think that it needed for a PCI parallel port card but it knew that it needed for a USB to parallel adapter. I only wish that I knew precisely what driver that it was. Maybe this will provide some clue so that the people who know Windows 7 better might be able to figure it out. I'm more than willing to answer questions if someone wants to figure out what's actually happened here.
For the record, the sequence which I went through was: I tried the printer (Installed using the manufacturer drivers for Windows 7, 32 bit) installed to the LPT3 device which Windows 7 recognised the parallel port card as. Verified symptoms as I laid out earlier. Print job appears to complete, but the printer does nothing. Uninstalled printer. Plugged in USB to parallel adapter, waited for Windows 7 to recognise the device and install drivers. Plugged it into the printer (No PnP response.) Installed the printer using the same drivers as before, but using the USB001 device. Printer worked. Unplugged the printer from the USB adapter and plugged it back into the parallel port card. Changed the printer to use the LPT3 device under the Ports configuration tab. Printed. This time it worked.
For repeatability, I did the same thing with the other PCI parallel port card and a similar printer on another identical workstation. This install started with installing the PCI parallel port card first, then followed the same sequence, with the same results.
So, my problem is solved, but I think that it's unreasonable to plug in a USB adapter to make a PCI card work.