I don't have a direct solution for you, but do have some tips. Some of the older HP optical drives did not have drivers on the bootable install DVDs for them, and others did. Some had drivers on the HP-supplied restore disc/partition, but not on the Microsoft install CD/DVD. If you got lucky you never saw this issue, but a workaround was to flash update the firmware of the drive (if HP had an updater available) and that would usually take care of things. I even saw this issue on a Z620's HP optical drive that could not run the W7Pro64 "system builder" DVD bought straight from Microsoft.
I had one type of older HP optical drive that did not have a firmware updater from HP, but did from Dell. I learned how to "cross flash" the firmware and the drive then ran with the Dell firmware and finally could be used with Windows 7. It could not with the HP firmware, but that optical drive did work fine on XP. Cross flashing was quite the exercise.
There is no HP FW update that I can see for your optical drive, but Lenovo and Dell do have one for the same drive. It might be worth looking into, and I'd go with the Dell firmware for cross flashing, but it really is not worth the effort you'll have to put in learning about that. My advice is different:
Go on eBay and find a "used" HP optical drive and buy that. The later versions have higher part numbers, as you'll see. Get something that starts with a 6 or even better a 7. You'd want a CD R/W and also it should have DVD R/W. You can get those, with shipping, for about 12.00 USD. Used means virtually new, usually, given how little these optical drives are fired up these days.