can my parents see permanently deleted messages on my iPhone, even if the messages are permanently gone?

As noted, unless your dad has access to your iCloud account, he can’t see anything on your iPhone unless he is looking at your iPhone. He can’t see any messages or pictures whether previously deleted or not. The ATT account only lets him see your call, text and data usage logs. 

Dad is gaslighting you. But you might want to change your iCloud password just in case he knows it. If he has your iCloud access he can get anything that's synced and backed up and goes through your Apple ID.