This is what I think is fair when one person works FT & one PT:
The PT person does as much as they can chores-wise while the otehr person is at work.
In my case this means when I finish work I pick DS up from school, then as soon as I get home I do the laundry, ironing, wash up breakfast things, go shopping if necessary, prepare dinner, odd bits of housework, plus all the things DS needs doing - snacks, hearing reading books, finding stray lego bits, doing playdough, etc etc
This means by the time H gets home from work all that needs doing is washing up & putting DS to bed.
The most H has to do is the odd bit of housework at weekends, mow the lawn & put the bins out.
So, if you are working FT, that is all you should be doing.
If he can;t manage it, tell him to get out there and find a FT job.