At first I thought such a thing did not exist, and as has been said, it probably doesn't under the MRA label. However, I just had an idea of something else you could try searching. In my Women and Gender class last semester, a guy came in to talk about a small local organization he started as a place for stay-at-home dads to get together and talk about difficulties in being a stay-at-home dad and fatherhood in general. It seems that they mostly address the feelings of lesser worth from being primary caregivers, the issue of divorce and custody, and gender stereotypes that they were personally hurt by. What's interesting, though, is that he said his group identifies as feminist, not masculist, despite the fact that they primarily discuss the specific effects of sexism that hurt men. I highly doubt you'll find any truly equality-minded men's rights advocates who are not also feminists.
You may have some success in searching for "equalist" forums, which is sometimes used by people who want to discuss sexism issues without talking about women specifically, but as might be expected, sometimes "equalists" are just as thuggish as men's rights activists because they want to discuss discuss sexism issues with even acknowledging women.
The percentage you're paying is too high priced
While you're living beyond all your means
And the man in the suit has just bought a new car
From the profit he's made on your dreams