Unless I've misunderstood, I think they're two different women - the woman who spoke about it ruining everything was from the cat re-homing service, not the OP's mum's friend who's trying to get a new home for her cat.
I think the OP has the right idea - for a start, she's asking for advice instead of just grabbing the first one she can get hold of. As for bringing a cat into a household where someone doesn't like them, I think her dad's just wary rather than having an outright dislike of them. Neither of my parents were mad about cats & my Dad still isn't, but when we got our first as children it was my Dad who had her on his lap more than anyone else, making out he didn't really care she was there but telling us to leave her alone if we tried to take her.

Cats know the difference between dislike/hatred masked by a smile & aloofness masking a sneaking love.
OP, cats are hard work & the younger they are the harder it is. A damaged cat is harder to deal with whatever age it is. Any cat coming to live with you will have to get used to a lot of new things - new people, a new home, new surroundings, new ways. Don't be under any illusions about scratching - cats scratch sometimes just because they can, & if it's not furniture it's the nearest available leg or your shoes/boots. Your parents & you need to understand that will be the case, & if they're concerned about this happening, then maybe getting a cat isn't a good idea. That's just the way cats are - a law unto themselves. Having said all that, if your parents are agreeable (& I mean genuinely so, not just giving in to your pleading), then go ahead. It'll take time for you to get used to having a cat too, but as long as you stick to your end of the deal in terms of looking after it I think you'll be fine.
I don't think this woman's cat is for you though. If she has other cats, she'd know the position re her neighbours & if they objected to her cats, the objection would be about all of them not just a particular one. I'd be surprised at any cat owner getting rid of just one of their cats without a better reason than a neighbour's dislike, so I think this woman's not telling the whole story. I'd let her sort her problems with that cat herself instead of passing the buck to your household. If (I suspect it will be more a case of "when") this cat does whatever it's been doing in her home in
your home, your parents won't be happy with it & may find that very difficult to live with. It's not fair to the cat or the rest of the family to end up in that position, when they didn't know about those problems when they took the cat in. In addition to possibly having the stress/distress of trying to find it a new home, your parents are also likely to be so annoyed that your chances of ever getting a cat again while you live under their roof will be nil.