I believe I know exactly where your parents are coming from. I suppose this is due to racial history outside of American borders, going back to the time when Korea had forcibly become a part of the Japanese empire. They probably feel very mistrusting towards her and her family because of what happened during that time. But if that isn't the case then I apologize for being presumptuous.
But what I will tell you is that like what most of everyone has said, you should not allow their feelings to get in the way of your relationship. Of course, I wouldn't dismiss their feelings outright as something foolish and distasteful for they're probably going through something. I can't say what it is, but as much as this is an annoyance for you, this is painful for them.
Time, commitment, and understanding on both ends could help to heal the wounds gradually, but that isn't a guarantee. I'd imagine that a relationship between someone of Korean heritage and someone who was a part of the African Diaspora might receive similar disapproval just because of past racial history.
We should never think that the past is dead and gone just because those of the past are no longer with us (or simply because the past is just that, the past). Forgetfulness, willful ignorance, re-definition . . . often lead to repetition. It's not a matter of equating one group's suffering with another, because they are entirely different and do not share the same social and cultural dynamics. It's about understanding why things are the way they are and trying to find a solution.
Sometimes the best solutions are find with proper study of the past.
I don't want you to abandon your family because in many situations family is all you have. I understand that your girlfriend and yourself really love and care for each other, but there is always a way other than to give up on those that don't seem to understand. You just have to be willing to try, because I'd find it very difficult to see such a young man running away to be with someone he loved. (I'm assuming that you're younger than 21, which could be wrong.)
It doesn't always work that way. Sometimes people make mistakes, sometimes you find out that the grass isn't greener on the other side, and sometimes you realize that they won't ever change and you'll just have to live with that.
And if they won't change, you may find it easier to love from a distance.