Is it acceptable for a professor to ask post-doc student to babysit for free?
Your friend's advisor has made a very unethical request because, as a supervisor, it makes it harder for the postdocs to feel like they can say no. And they should feel free to say no, since the request is not normally part of any university employment contract I've seen. 

Now, if the request were at the workplace, incidental and of brief duration (something like "Could you watch her while I take this call from the doctor's office?", for instance), then it wouldn't be so problematic (although still less than ideal). But anything more than that—anything that involved a regular arrangement, or was of extended duration—should be handled as a separate business transaction, so as to avoid exactly the coercion problem that you've raised.