Ahh, you see, I've always thought the other way around - if you're allowing me to look after YOUR child: I need to realise that they aren't my children but yours. I mean, obviously, I know that- but my point is that something you might just do with your child - like handing out paracetamol - is not something you'd do with another child.
They're not my child so I'd be checking it's ok first. Food, medical conditions? Check if there's anything I need to know before looking after them.
That would give me an idea of if we needed an inhaler or to consider foods. If a situation arose whereby paracetamol or ibuprofen were needed, then unless the parents of the child had previously (as in that visit) said it was ok. I'd be calling to explain the issue.
If I left my child with you - I'd be telling you everything you need to know and expect you to call me if a situation arose.
The only time this would ever change, for me, is if that medicine was lifesaving. For example, a severe and obvious allergic reaction where antihistamine was not anticipated and discussed but quite clearly needed. Go for it, please. 100%. Lifesaving treatment? Yes. Don't delay. Please.
But - don't just be handing out medicines at any other time. It's the way respect and trust is built and can be easily broken.