It's extremely difficult to gain admittance to a place like Stanford if your parents are not supportive of your educational ambitions. This is both because your academic results earlier in life (upon which admittance would be based) will tend to reflect an unsupportive family, and a place like Stanford wants students with supportive families. That's why parental support matters.
Of course they do, but most won't get it. It's one of life's many crappy realities. There's a line from a song..."Where it ends...usually depends on where you start". There are of course exceptions, but this generally holds true.
Or, maybe Stanford counts on parential guilt? They have quite a big savings account. (Stanford’s $21.4 billion endowment (as of Aug. 31, 2014), and grateful donors give the school close to a billion dollars a year. I don't know how many full scholarships Stanford gives away per year, but I hope it's a lot?