I think that if your husband is doing all the life admin, and you're doing none, then it completely depends on what you see as life admin and how much of it there is?
I do none of the cooking or cleaning in our house, but I do sort appointments for 4 people, organise fun trips to a T because we're a multi-disability family, liaise with school and make sure I'm always on hand, sort all the paperwork out that comes with being a multi-disability household, research topics for advocacy where needed, sort out when my elderly relative gets stuck in scam situations, spend my time for free looking for cheaper solutions and workarounds to things they want/need, DIY, garden maintenance, physio of our long term foster dog, organise the food shop and shopping lists which again need to be done meticulously multiple times a week due to disabilities, etc.
It's a lot of unseen and unpaid work, and I am exhausted by it.
Even if your house doesn't have additional needs to cater for and are a typical healthy family, there's still a lot of admin that goes on behind the scenes, and if your husband is doing that on top of working and you're not having to do any of that, then I can see why he would feel it's unfair for you to go to work, and then complain that you want to share your household jobs with him too, unless you were willing to take some things off his plate and share the load.
I'm usually the first person to call lazy husbands out, and sexist ideologies, but it sounds like you're quite heavily supported by your husband, and he's just set a boundary that if you want to do this, then it can't be at the expense of your responsibilities to your primary family.
If your parents are no longer financially savvy, it would be better for you to request a financial POA and take over their finances for them if you want to support them, but only give what you can reasonably afford of your time and energy.