It takes a big person to admit when they can't handle things. But doing so and then getting help is a necessary thing if you want to take care of those things and not be buried by them. That give and take is how to stay in balance with your partner.
But it's not easy...
The biggest problems in any relationship happen when it's out of balance. One person in the relationship is doing more to hold it up than the other.
Now I'm not advocating codependence here - far from it - but there has to be a balance in any relationship for it to work.
What should be looked for here is"What can my partner do to help me?" not "What do I need from my partner?" because that is a different thing.
In my opinion people should, within the limits of biology (no arguments about who can give birth ok?), be able to function as a single parent if necessary. That means that when you aren't a single parent you both know the duties involved and can perform any of them on a regular basis.
Sometimes there are reasons for not doing a task but you should do something else if you can't do the task in question.
For instance I refuse to give my daughter a bath. Because my daughter is 10 years old I don't think it's appropriate for a number of different reasons.
So instead I do a buttload of other chores that could be done by my wife to even it out a bit. (Just a note here - my wife and I are not in balance so my view is a little biased. Just FYI.)
In the end I suppose it works out but there are often times where one partner is working their ass off to maintain certain things while the other one is sitting on their ass.
Well you have a few choices in those cases:
1) Continue to work and hope that it will even out down the road.
2) Communicate with your partner how you feel and work out how you two can bring the situation back into balance.
3) Leave the relationship.
That last one is a true last resort because it not only breaks the bonds with your partner but it weakens the bonds with your child in most cases.
I personally pick the second one but sometimes that doesn't work. Often I'll follow these in increasing order from first to last. Thankfully I've never had to follow the last one...
Now remember that "What can my partner do to help me?" I mentioned earlier? You'll never get that by going that direction.
What you need to do it come at it from the other direction - "What can I do for my partner?". If you do that and communicate with your partner so that both of you think that way then things should even out pretty quickly.
And if they don't...well there's always option number three...