That sounds like a really frustrating situation, and I'm sorry. It seems to me (and I'm doing this "diagnosis" purely from what you've typed, obviously. I haven't seen or interacted with your dog, and I only play a dog expert on the Internet, so if I'm way off with any of this, no harm no foul).
Fear barking or stress barking are not necessarily indicative of abuse. It's possible she came from poor breeding, and was then poorly socialized for all of her life, and there's a laundry list of people she does not know how to react to.
What does she do other than (or in addition to) barking around you? Will she eat or take treats? Will she ever settle down on her own, and if so, is it in the same room as you? What kind of exercise does she get? What do your parents do when she barks? Has any training been done with her, and of what sort? (clicker training is something I love and will always recommend, though it does take patience)
In my experience, food bribery doesn't always work. Taking the dog out for walks and playing games he enjoys builds trust and is team work, plus something you may actually enjoy as well.
I think this is a fantastic suggestion!
There is really no way to say what the dogs motivation is based purely on barking at one person and not others. But pairing yourself with good things like treats is not going to hurt regardless.
Bold mine.
Also this. Making yourself a source of the positive can result in a lessening of the barking behavior. I read a blog post which I've never been able to find again where there were treats out at an animal shelter, and a sign that said something along the lines of "Give treats even if barking." Dogs can't bark if they're eating. If humans mean only positive things to dogs, their stress levels can be reduced (especially if the barking is a stress/fear reaction) and other behaviors can be developed.
You might consider what you'd like her to do instead of barking."No", which means a whole lot to us humans, doesn't mean a whole lot to dogs (it doesn't mean I'm not guilty of using it, but there's the added Fun! Wow! factor that if we're yelling too, that's reinforcing to the dog.), and it is not in itself a command. I mean, it kind of is, meaning "stop that", but it does not indicate what should be done instead. "Go lay down" may be a good interruptor, "find your ball", that kind of thing.
If your mom got this dog before Christmas, she's only been in the family for a few weeks. That can be overwhelming for a dog, new house, new smells, new routines, new people. It could be that, by this point, you haven't even seen her "real" personality just yet. I've seen people who've been in dogs for many many years talk about amounts of time it takes for a dog to get comfortable in a new space.