Do they know you've been together for ten months, or do they think it is new?
Personally, I would not stop things with the girlfriend. I would tell my children "she is an important person in my life, and I don't abandon important people in my life. Not you, not your mother, not my girlfriend".
If you just dismissed her when she's not convenient, I feel it would give the message that you might very well do the same thing with your wife or your children.
Other than that, just like when parents learn and react badly, I think the only cure is time. Ultimately, they don't have a say in what happens in your love life, although I do think they deserve to know if the girlfriend is going to be over. Just stay open to answering any question they might have, reassure them you love them and their mother just as much as before, and don't be the one to bring up the subject.
Family counseling does sound like a good idea too.