Elephants I think you may have put your finger on it. IceBeing, I feel for you, on the one hand it's quite trivial - washing-up, for goodness' sake - but on the other hand it's revealing of deeper issues. There's always so much more to these things than you can easily put on a MN thread.
The elephant in the room is that you had PND, how crap that must've been, and your DH kept the boat afloat one way or another while that was all going on, so he can't be a total bastard or eejit. And it was obviously good for DD that he was around lots during the early years, I wish I could've been with my lot.
But time spent looking after a 3yo girl isn't coal mining. I bet she's delightful - 3 is such a great age - so it wouldn't be fair of him, in any discussion of workload, to balance that off against true drudgery like washing crusty dishes. Sigh, I am very anti-dishwasher, I think they're a snare and a delusion, but I'm beginning to think it might work for you. Not that you can do crusty pans/bakeware in them anyway, damn them. The real reason I don't like them? They totally fudge the issue of who washes up, and the dishes are never put away properly!
But Elephants makes sense; it sounds as if your DH somehow thinks he's doing you all a favour by being a SAHP - and maybe you've agreed, or acquiesced? Not unreasonably, because you were going through a really stressy time and having him at home really did help?
In our house, my new-man offer to put my 'career' on hold to allow DW to go full steam ahead with hers (& she earned more per hour) was met with a loud laugh & the reply, 'Thank you darling, but if anyone's gonna do less work in this house, it's gonna be me!' And when my grievously wounded feelings recovered, I could see she was in the right. She already did more housework, and staying at home more, all other things being equal, is a prize - although paid for by shouldering more chores. Most of which aren't so terrible anyway, I really like pegging out big loads of bright, cheerful washing on a sunny day. I've found chores are only really irritating if you feel they're distracting you from stuff you'd rather be doing. So maybe DH is actually not 100% happy being SAHP any more, and needs to expand that hobby into a job?