It's a known problem. Here's why. The timing of the newer machine's parallel port in much faster all around since not only is the circuitry faster, but your CPU is likely faster.
A known fix is to add a PCI PARALLEL PORT CARD.
I'm surprised Dell didn't tell you this (or Iomega).
The issue of the drivers confuses the owner and gets them to think it's purely a software issue. Sadly I can't fix that.
In closing, the ZIP 100 Meg drives are very expensive media wise. I have removed all but a few from the office and moved to CDRW. The last units are IDE ZIPS which are not as finicky as the parallel port model.
Have a new Dell 8300 with OS XP home. Connected parallel printer to zip drive & zip drive to parallel port on computer. Downloaded Iomega's file for XP zip drive & printer were recognized & working. So what's the problem you ask... Well, now at cold boot the zip drive is not recognized (& the printer doesn't work), i.e. if I look in "my computer" zip drive is not there. If I reinstall Iomega's file everything fine & both work until the next cold boot.