Wow nothing like this has ever happened to me but I think I would flat out ask the Dad as Sunnydays suggested. There is no point enduring this and causing stress. If they are not happy with your services then they can go. You also deserve respect and maybe just talking will clear the air and make the dad realize that his behaviour is childish. If that's not possible then you can stat advertising and when you have someone then ... you can let them know that you are letting them go as you sensed they were not happy with the services and remarks from dad made you feel very uncomfortable and unappreciated and that they may find care that suited them elsewhere.

I've never been terminated but I have before. It was the worst feeling ever but I had to do it has this family had hygiene issues that were not addressed after multiple warnings and I feared for the health and safety of the daycare. Anything making me feel stressed out or anything negative that can not be resolved any other way but by terminating ... well that's what I'll do even if I really really hate it. I CHOSE this job to be happy so ... why endure ?

Good luck