This family don't sound like they agree to what you outlined above plus different arrangements will work for different people. Also, if mum has a babysitter watch the child then yes that is her choice but the minute dad says he isn't happy with the person caring for said child, then she should really respect that as he has the right to decide who his child is around too. This is where it is a joint responsibility and while there is some wiggle room for independent choices per parent, should the other disagree then it should be taken into account. Also remember they are amidst a divorce so likely nothing has been agreed too. I know as a parent that I would not be happy having my child left alone with another caregiver that I did not know. This sounds more like them being asshats with each other but we can not begin to know exactly what's going on behind the scenes.