I have done this as a parent. I would have the day off but book myself a physical or have errands to run or decide to steam clean the carpets in my house so I would still send my child(ren) to their daycare....or I would go have a nice lunch with my husband just the two of us and take in a movie with him and have a date.
I didn't want the daycare to judge me so I would still dress as if I was going to work and drop off as usual.
BUT I would usually go pick up a couple of hours early and use the chance to do a surprise arrival and pickup. That way I still got some extra hours with my child, got to check on the provider too. I would tell the provider I got off early or something.

Of course this would be just a one day or two day a year thing. There were other days where i would have vacation or book days off to be with my kids and just spend a good day with my kids. I would let my provider know and they would still get paid for the day of course. It was worth it. I especially always made sure to take off my kids' birthdays so I could be with them and have a mommy and me day.

I can see why people would need some time to themselves to do errands etc

But now that I am a provider --it has been interesting to flip the coin and see parents blatantly tell me they are off and then still drop off and pick up at the very end of the day. Once I even had two parents have the day off...tell me they are off...and still drop the child off on their birthday just so they could get ready for a party...or parents who still come right at 5 when they have been off all day....but what can you do?