I've just gone back to work 3 days a week. DD is nearly 3 so I had a good stint as a SAHM.
We agreed, before she was born, that is just as important for the person looking after your child to be well rested as it is for someone doing any other job. So:
While I was BF at night, I did nights but he did the mornings from 4/5, so I got a bit of a lie in.
Since she stopped feeding at night, we've done a night on, a night off. Whoever is up in the night, the other person does the morning.
If he has a particularly challenging day ahead, I'll take an extra night shift BUT, he still takes the morning and he packs the nappy bag, dresses DD and does her teeth and hair so I can get out of the house more easily the next day.
He had 5 weeks pat leave and changed all nappies in that time. If he's off work or WFH but not busy, he still does them. (I don't ask but if he hears me talking to DD about changing her happy or using the potty, he'll often do it.)
We take bed and bath times in turns.
I probably end up doing more laundry and cooking. The former is easy to do with DD around and the later needs doing while DH is working because of DD's meal times. Everything else is split 50:50.
Maybe if DD was a less engaging kid I'd do more. But the way we see it, my "job" is to interact with her. If we were paying someone to look after her, we'd be furious if they were just doing household chores. Plus, anything I do, she wants to join in. It's good for teaching her household management, but it all needs to be redone after!