Can I use my Mac to backup both my iPhone and my dads iPhone, and then have our information separately transferred to our new phones?

Yes you can, just make sure to select ''set up as new iPhone'' when you are making a backup of a second iPhone on your Mac. What you are saying to the Mac is to see this as a new iPhone and not like the iPhone that is already backed up on your Mac. Apple has some video's to help you out, one also explains that you do not need a backup to transfer your data (works on devices with iOS 12.4 or later os iPhone 5s and later if up to date). Here are both video's:

https://www.youtube.com/watch?v=JNSbVc11Yn4

https://www.youtube.com/watch?v=cpNk6fkwS2I