I will disagree with Melange. It is not necessary to attend their church to show that you respect their beliefs. In this situation, your best option is to take the direct approach.
With the secular movement growing in the US and the world at large, many Christians feel like they are "on the ropes", as it were. Christians feel very threatened by atheists and secular humanists right now, so dating a Christian, especially an underage Christian (I presume you are underage since you're worried about the family, if she were an adult, you would not need his permission). So your best bet is to just go visit them, sit down and have a nice talk with them about their family, about their beliefs and about their rules. Remember, if you have any ideas about being with this girl, it is important to get to know her family as well. Christians respect family-oriented values.
As you talk to them about these important issues, bring it up to them directly that you like their daughter very much and you would love to date their daughter, relate to any values you share with them and talk about how you would respect their family values, but let them know that you're an atheist, but that doesn't mean that your values are very different from their own, and you wouldn't want them to feel threatened by your worldview; you're not trying to deconvert their daughter or bring some kind of demons into their house.
If they accept you at that point, then you're fine, if they reject you, you can try dating her behind their backs, but that will cause trouble between her and her family. Ultimately she will make her own choices, but until she is a legal adult (provided she is actually underage as I assumed before) then she has to run things by her parents, who are legally responsible for her.
Most importantly, you must be willing to accept that you may not get to date her at all. You must prepare yourself for that possibility, or you will not be viewing the situation in realistic terms. Be direct, be respectful, be patient, and most of all, be yourself. Don't put on an act. Be on your best behavior, of course, but don't pretend to share values that you do not. If nothing else, you will earn respect for being honest, even if they disagree with you.
If you'd like to discuss this further, as I'm always excited to link up with other atheists, my inbox is always open. Message me anytime.
