I don't think that's common at all. It makes no sense to me. I'm an MB and we always pay the same rate. We pay for supervision and adult presence in the house.

Nanny here - it's not uncommon for parents to ask to pay you less while the child sleeps, but it's wrong.
Save yourself the stress, and don't work for families that want to nickel and dime their childcare.
I assume parents who propose to pay a nanny half, or less than half, time while their child(ren) sleep expect the care taker to do his or her entire job if the child(ren) wake up ill, with a nightmare, or choking.

Not during naps but I do other kid stuff while thy nap 2-4 hours a day. Like cook, clean, meal prep, laundry, etc.