Save yourself frustration and give up on the USB->Parallel converter. In my experience, they suck and hardly ever perform bi-directional communication properly.
A lack of bi-directional communication may very well be why the printer is failing when trying to retrieve the (extended) printer properties, and erroneously reports the driver isn't installed properly, instead of an actual communications error.
HP's full client drivers often expect, and depend on, full communication with the printer to provide information like ink statuses, available printer features (Collating for example) and such.
Since it's a desktop computer, consider buying a PCIe parallel card (~$50), or possibly a new USB-native printer.
The cheaper alternative is probably to revert to a basic, network friendly driver that's missing all these 'extra' features. That's probably why the HP Universal Print Driver was being used in the first place, as it's much more forgiving when bi-directional communication isn't available.