Both I guess.
Honestly, as "moderate" as they are, I don't think they will. My mom really stresses that she would prefer me being with a Muslim-born man than a convert, but my dad doesn't care for either, as long as he's Muslim. Even if I had a guy who was willing to convert to marry me, I don't think my mom (and probably my dad, can't say) would be satisfied. They would rather him be more serious into converting for the religion and not just for marriage.
To be honest this subject bothers me a lot when I think about my future. I'm not gonna lie, it sorta scares me too.
Hah, you're lucky that you got away with it at least. Most Muslim families do that, just brush aside girls that their sons date. However most Muslim girls get into much more serious situations if they were caught with a guy.