My parents have separate apple IDs but my mom’s ID uses my dad’s email and my dad’s ID uses my mom email. Can I remove one and add it to the other? Or will removing one make so I can’t use that email again?

The short answer is, based on what many other users have reported I think you are correct about not being able to reuse the email addresses.

Although this next section discusses email address re-use in the context of deleting an account, it may provide general guidelines as to not being able to re-use an email address once used as an Apple ID.

From: How to delete your Apple ID account - Apple Support

"Can I use the primary email address for my deleted Apple ID account to create a new Apple ID or with another existing Apple ID account?

No, once your Apple ID is deleted, you can't use the email address that you used with it to create a new Apple ID. In addition, you can't use that email address with another existing Apple ID except as a rescue email."