OP, what a muddle.
I still don't really get why your DH (you refer to him as DH, but are you actually married?) is doing his own washing and making his own meals. Why isn't he sometimes doing both, for both of you, and you likewise?
There's too much to unscramble here, really.
The C-section thing is horrible.
However, it's easy for people on here to brandish their pitchforks. The fact is that you are likely to be tied to this man for a very, very long time as a result of having a child with him. My XH was diabolical, but there is nothing that could have persuaded me to be without the children for even a single night when they were small.
Your ultimatum is evidently the result of understandable frustration. What's more, you're stuck with it now that you've made it. No, he won't agree to take over all responsibility for the baby for a week. So you now have to go to your parents' house.
That in itself might not be a bad idea, if it's a week's 'holiday' there, or even a fortnight's holiday. It would give you time to rest and think more clearly.
However, I think you also have to bear in mind that for most men, the idea of sitting together on the sofa talking about feelings is about as appealing as having your teeth pulled out without an anaesthetic. Fifa is going to seem far more appealing than that. Men also often find it difficult to have a partner who's endlessly moaning (in their view) about being lonely, miserable, etc. Men are hard-wired to solve problems, and many of them dislike being confronted with a woman who just wants to talk, without necessarily wanting a concrete and immediately realisable solution.
Obviously not all men are like this - but your husband sounds as if he is one of those types.
As I say, a muddle.