I wouldn't do it... and I don't see how a parent could manage it without my consenting. I am confident in my care, but I completely agree that things could be taken out of context. I would not want to feel that every little thing I am doing could be called into question.
I also agree that if a parent cannot feel confident in the care, it is not the right care situation. And if they don't feel comfortable in ANY care situation, then working outside the home/ not having 'in-home' care is not the right thing for them.