I don't claim to know everything about team parenting, but I can tell you what doesn't work: keeping score. If you're trying to get your partner to take a bigger share of diaper changes or getting up with the newborn in the wee hours, the worst thing to do is start saying, "I did it this many times last week, now it's your turn." That's asking for endless disputes about who did what and what constitutes a "turn."
The thing about dads who come home and do the Hulk Hogan routine is that they're fiercely proud of it. It's one great thing they can do with the kids if they work all day, and they feel guilty about not being around more. I understand that even better now that I've been in both positions. Complaining that he only does the "fun" things with the kids is only going to hurt his feelings, if not make him outright angry. Would you rather him be the old school guy in a flannel suit who comes home, pours himself a scotch, sits down with the paper and says, "What's for dinner?"
Take advantage of what your partner is good at with the kids instead of insisting that it always be an exact 50/50 split. Besides, if he's not pulling his weight with the midnight feedings, you can always pretend you're asleep.
Bills, bills, bills