I think it’s important for you to separate the idea of being a dogmatic, “legalistic” Christian (it sounds like your parents fall under this category) from being a Christian in general.
At the end of the day, do you accept that you are, fundamentally, a sinner that is incapable of living a blameless life and accept that Christ died for your sins and that He needs to be at the helm of your life?
Encourage your child to bring the friend home for a visit.
Let the friend see a Christian family up close and personal. If the friend is uncomfortable and doesn't want to, be patient, and keep inviting. You don't have to withhold the truth, but there is a time when you must speak and then let things be. Avoid the temptation to engage in constant shaming of them, or reminding them of your anxiety over the situation.
I have been on both sides of this matter; I was the unbelieving girl who dated someone's son, and I've been the mother of a child who dated an unbeliever.
It does mean recognizing when we must allow our children to be the adults they are.
We can only control our responses, ones that are guided by grace, love, and kindness.
You should not marry a non-believer, and therefore dating one is a bad idea.
You are setting yourself up for a series of potentially damaging slip-ups. I lost my virginity to an earlier, non-Christian girlfriend.