I'm an atheist so just the idea makes me roll my eyes but I assume they don't like it because they think the other kid may influence you and may convert you to another religion or they are just prejudice against other religions.
14 Do not be unequally yoked together with unbelievers. For what fellowship has righteousness with lawlessness? And what communion has light with darkness? Also like lala said. Most have issues down the line and will eventually end up in a divorce because of it.
Religions are profit generating machines, and they are in competition for your money. If they can keep you in their religion, they get your money instead of the church across the street. Becoming close to people from another religion is a threat to your church, so your church makes people (including your parents) believe that its bad and against god to do that. Ideally, they need you to breed within the church membership.
I personally wouldn't care if my wife/ girlfriend was religious or not. As long as she would be open-minded and open to new ideas I wouldn't care. I guess many parents, both religious and non-religious, don't like not being in control of their child and they would probably see an act like that as defiance. Also this is coming from a die-hard Christian