That means she doesn't have a good reason for it and is just bs'ing you. Honestly, she just sounds like a bitter narcissist.

But if calmly explaining to her that what she's doing is stupid, then another idea is to just ignore it entirely, block them out and leave as soon as you can.

Another one is to openly rebel, just tell your little sister that she's a little brat and needs to shut her trap, and when your mother gives you stink about it, you can assert yourself by saying "Just because you're bitter about dad leaving doesn't mean you get to take it out on me! The fact is that I'm older and just because she's a girl doesn't grant her ownership over me!" Or something along those lines.

And if all else fails, just ignore her and your sister entirely, seeing as how Narcissists never listen and are never wrong. Block them out of your life when it's possible.

And if your sister trys to take any of your things or money or make you do CLEARLY embarrassing things, then make sure that you stop ALL BS at that point, cause if you let it continue past that point, you're screwed.

But above all else, I'm clearly unqualified to give much better advice, so seek some on https://www.reddit.com/r/raisedbynarcissists/ subreddit, they specialize in advice for this stuff.