Thank you so much for the help, Rick, I Am A Baka, and OoTmaster.
OoTmaster, I have a question.
Can you say the same thing about the following pair?
(a) This door doesn't open.
(b) This door won't open.
Hirashin
These questions are actually quite tricky, and the more I think about it, the less sure I am about the answer. To me, the main difference here is that "doesn't open" implies that it's usual for it not to open - either it was deliberately blocked for some reason, or you know it's been broken for a long time. "Won't open" somewhat implies you weren't aware of any reason why it shouldn't open, maybe the door is broken and you found out when you tried it. I don't think the usage is exclusive though, you could say it the other way round and it wouldn't be wrong, just less common. I could be wrong. I think this usage is different to "my parents won't" because it's an inanimate object. I don't get the same sense of usual/deliberate vs unexpected/accidental when you say "my parents..."
*edit. Actually, now that I think about it, if you only just found out that your parents won't let you work, you would say "won't", not "don't", so the usage is kind of the same as for the door... if the context was something like:
A: "Did you find out if you can take that part time job we talked about yesterday?"
B: "No, my parents won't let me."
B wouldn't say "my parents don't let me", because that would imply that B already knew that when he talked about it yesterday, and A would wonder why he didn't mention it at the time.
But I think if the context was entirely about the present/future:
A: Here's a good part time job. Can you take it?
B: My parents won't/don't let me work part time.
To me, "won't" and "don't" mean the same thing here.
Now I'm even more confused. LOL.