Here is a list of largely self-taught, highly successful people. I think there are many examples of people who have succeeded despite little or no involvement on the part of their parents. Of course it's not ideal, but it certainly is possible.
OK, I might have missed some, but I saw no classical musicians and no 20th century American mathematicians, with the possible exception of Walter Pitts. I don't know who he is, but it says that he was "one of the scientists who laid the foundations of cognitive sciences, artifical intelligence and cybernetics." Those are all new fields, so that kind of makes sense. Not to say it's not an example that refutes my statement - it certainly is.
I wasn't really thinking about historical figures because the educational avenues available to people even just 100 years ago were so different. I would be interested in hearing about someone's neighbor or cousin who obtained a PhD in math at a highly competitive university or works as a professional classical musician who attended school in the U.S. and did NOT have parents who encouraged and facilitated their achievements pretty proactively.
P.S. - Oh - my bad! Joachim Raff Georg Philipp Telemann are two self taught classical composers. One of the others listed (Edward Elgar) had a parent who a was a musician, though.
