I've come to the realisation that my next door neighbour is being abused by her parents (she still lives with them). I think she's in her late twenties/early thirties, I don't even know her name and we've been living next door to each other for at least sixteen years (I'm 22). I've never seen her with any friends or a partner, in fact I've never seen her without her mother. I know they work together but other than for that I've never seen her leave the house, at least not without her mother.
There's always been screaming and shouting in their house, its been going on so long and from when I was so young that I barely noticed it until I came back from university. Now I notice when she's crying outside in their garden, and I can hear her being hit (with something like a slipper or shoe?) through my bedroom wall while they shout.
I've asked other people in the village if they know anything about her, and it takes them a while to even remember she exists, which is absolutely shocking because I think they've been here at least 20 years. I don't think she has any friends, and its horrible hearing and seeing her upset and getting screamed at and hit and I don't have any grounds to just call her up and ask if she wants to come over for a cup of tea.
I'm wondering if maybe I should start just popping to their shop and buying something from them for my pets and striking up a conversation with her each time, and maybe stopping her when I see her in the garden and ask how her day's been, just to get something going so if I do see her crying it wouldn't be weird or none of my business to ask if she's okay and if she wants to come round.
But then I wonder if I should stay out of it, or if I should do something else, I doubt if I told anyone or called the police she'd say anything against her parents, she seems to have good and bad days with them, so she might not even realise its not okay. I don't know what to do.

OF COURSE you can ask her over for tea. She's your neighbor? Why would you think you 'should stay out of it?' If her mother shoos you away, then go to plan B, popping into their shop.
Sometimes a parent will terrorize a grown child just to keep them from being confident enough to leave. The daughter might have an emotional or intellectual handicap that was lifelong, not just from all the abuse, and the mother is clueless about how to handle it. I can't think of any other common reasons for all this, but hope it isn't something more sinister (this is sinister enough). You might be able to tell more by lingering cheerfully in the shop, asking dumb questions about pets. Let us know, will you?

She's an adult... she could always leave.

It sounds like abuse has been going on for years with what you have heard yourself. If she has never been allowed to have friends, or go anywhere without being escorted by her mother, I wonder how much success you would have in becoming a friend to her.

That being said, what if you were to become a confidant and learn what you already knew about her being abused. If she has lived a life up to her 30's being totally controlled through force, what would you do.

I can say that personally, if I were convinced that anyone- man, woman, child- were being physically abused, I have an obligation as a professional, to report this to the authorities. I don't know how you feel about this, but I'm pointing this out, as It is a possible solution to help for this woman.

Because she is close to 30, the police knocking on her door can only offer assistance if she wants it. The assumption is made that she is legally capable, as an adult to make her own decisions, and with her having lived this way for so long, she may be unwilling, or unable, to take a leap of faith into the outside world, and find the resources she needs to survive without her parents. And who knows- perhaps her parents are guardians because she has some serious problems, psychologically or physically.

If you were able to become her friend, it would be an opportunity to provide her with information on women's shelters, which would be able to assist her in many ways, to become independent. That would be the best possible outcome.

Being a friend IF she is willing and able is a good thing, but it's a fine line between wanting to help and being able too. I like your idea of LEARNING more, just by being a friendly neighbor, but don't go snooping around or intruding in the lives of others, uninvited with helpful intentions, without a plan or commitment to follow through. I say that because in your other post you have plans of your own for the future, and given the fact you may be very limited in being able to actually be there as the kind of friend who can actually be helpful.

Don't start something you don't intend to finish the right way. I like your heart though, but temper your actions with good sense since this is the life and welfare of a stranger you are dealing with. You really don't know how to help her to be frank, or what help she really needs. So if you don't have the time to get ALL the facts, just reach out as a good neighbor, and leave it at that. More facts can guide you better than just feelings to help.