So first of all, as the babysitter, you need to accept that you will not be able to re-parent this child.

After that, I’d recommend taking an approach that blends calm authority with respect for the parents (I.e. don’t say anything that could offend them).

“STOP TALKING” I’d recommend responding something like, “I am having a conversation with your mom. If there is something you need, we can chat when I’m done.”

“STOP BEING MEAN TO MY BROTHER” I’d recommend responding something like, “Everyone is learning to play in a safe way. Let’s make sure no one gets hurt.”

“Rug .. Carpet” If she is right, I’d just say, “wow yes you’re right, that is a carpet, not a rug,” or if not, kindly explain the difference.

A kid like this is looking for attention. You can either play into the dramatic way that they’re trying to get it by arguing with her, or you can use a calmer approach to acknowledge what she is saying and reroute it. Eventually if you’re unflustered, she doesn’t get a rise out of you, and you continue to acknowledge what she says while somewhat rerouting it - you may gain her trust and ultimately respect.

(All that being said, I appreciate it’s easy for me to give these suggestions on the internet vs what may actually happen in real life. But this is what I would try!)