Everyone has their own individual mental load which is, as you say, life admin. Remembering to tax the car, insure the house, renew the passport, book the dentist etc etc etc.
If you have a child, to a certain extent, you now carry their mental load too. Lots of this is non negotiable - child's doctor appointments, dentist, applying for their school place, paying subs for their clubs, buying & washing their school uniforms and other clothes. Etc. Life stuff, but now more, for the child and any following children.
I'll be honest, some of this is optional and some parents do choose to carry too much of their child's mental load. For example, it's your responsibility to purchase them a PE kit and wash it when it's put in the basket. But it's not your job to go digging through their bag to find it etc at least certainly not for an older child. Nor to remember when their homework is due. You can adapt how much of this you choose to carry, but as they grow, there remains some non-negotiable items.
And then there is a partner (if you have one). What should happen is they retain their own load, and take on some of their child's alongside you.
What actually happens is the woman seems to take all the child's, her own, plus her partner's. This is where you find women fretting about renewing licenses for their adult partners (who would have had to do it themselves without her stepping in), scheduling appointments for her husband's health, checking when the tax runs out on his vehicle etc etc.
This part is entirely optional, there is no non negotiable part of your partner's mental load you should be carrying. Let them feel the consequences is my personal feeling on this stuff.