I don't want to invite my parents to my wedding because I know they don't approve of our relationship.
Well, what about your relationship do they not approve of exactly?
It turns out my girlfriend is half Japanese.
It's reasonable that you don't want to invite your parents to your wedding if they don't like your relationship, because the day is about you and your girlfriend. You don't have to include anyone in the wedding that makes you uncomfortable.
Really?
Yes, it is completely find to not invite your parents if they want to ruin a day meant for you and your fiancee. You could try speaking with them first, and maybe they would agree to be on their best behavior. Otherwise, you are making the right choice.