Apple ID does not exist

My father set up my iPhone with his Apple ID a while ago, and I've always been logged in. I just upgraded to a new phone, and I restored a back-up on my old one, with this account. Apparently, he had to create a new Apple ID, and the old one hasn't been used in over a year. My new phone keeps asking me for the password for the old account, but whenever I try to reset the password, it tells me the ID doesn't exist.

I contacted Apple support, and they sent me an email to reset my password, but it only reset the new password, not the one my phone is begging for. How do I remove the old account from my device, if it doesn't have a password to remove it?