I think you have to do what works and what both parties work out together.
In our relationship we view the SAHM part as taking care of my DS. It does not include housework. So my "job" is to take care of Astons needs and I also have a part time job where as my DH has a full time job.
So anything to do with DS is like 80% me. I 95% of the time have his bags ready, get him up and ready and do breakfast in the morning as DH is already left for work. Bedtime is a joint effort.
I also take care of the logistics of the family - budgeting, schedules, appointments etc.
Housework is whoever can do it and works out about 50%. Cooking is whoever feels like it and the other takes care of DS. Clothes washing will just be whoever can get to it. Dishes is mainly me as I try to do it during the day. I mainly put toys away. DH does 100% of mowing and most outdoor stuff.
DH does his own lunches and I do mine and DS or sometimes he does ours and vice versa.
I do the ironing, dusting and bathrooms.
We have a robot vacuum so not much of a task. I do grocery shopping.
Also if one of us is down for any reason the other will pick up the slack.
I think if one party is not happy with the arrangements there needs to be a discussion and both parties needs to work together to figure out what will work.