I don't think there's much for you to do or say at this point. Things like this don't magically heal with the right words. When I got sober I wanted everyone to forgive me right away. I knew things were different, so why couldn't everyone else see that too? I was the only one who knew that my promises meant something this time around.
I think it's highly unlikely that they never knew about your alcoholism before the total breakdown of your relationship. Even if they didn't see it, if it was that bad your gf probably said something about it. I do think you are lucky that your gf has parents who have basically said they're willing to give you a second chance after your past history with their daughter. Most people wouldn't. I probably wouldn't.
It's probably just going to be uncomfortable for a while, until and unless they see a lasting change in her life. That's not a bad start, though. Give it some time, remember that actions speak louder than words... if everything is really all better now like you said then you should be fine, right? What's the rush?