It is true that below some income level many schools will expect no parental contribution, but the $125k / year level announced by Stanford is particularly high. Filling out the calculator at the education department(1) it appears that the FASFA expected family contribution for that income level would be somewhere around $20k (+/- depending on things like assets and family size).
Harvard, for example, makes that guarantee only as to students whose parents' income is up to $65,000 a year (2)
Edit: In light of dollaaron's correction below, this is much less significant than I thought at first glance. This part "Scholarship or grant funds will be provided to cover these costs in lieu of a parental contribution" is still above and beyond what many schools do, but is similar to Princeton and Harvard, as he points out.
This is a no tuition guarantee, not a no parental contribution guarantee. In effect they are saying that if you make under 125,000, you'll only pay 20-25k for living expenses (room/board/textbooks/etc), which is similar to Harvard and Princeton, considering they both try to structure aid so families do not have to take loans out to pay them. Coincidentally (or not), this is also equivalent to the FAFSA EFC as you mentioned.
In this announcement Stanford clearly says that the no parental contribution level is 65,000, raised from 60,000 and now equivalent to Harvard/Princeton.
The "parental contribution level" was always frustrating to me. There's no way my parents were going to pay for my education, so that number being high and tuition being zero doesn't really help anything.
Am I misunderstanding how that works, or are they just saying "the parents should pay x and the student z - x" when they really mean "the total cost is z?"
Harvard, for example, makes that guarantee only as to students whose parents' income is up to $65,000 a year (2)
1 https://fafsa.ed.gov/FAFSA/app/f4cForm
2 https://college.harvard.edu/node/426
Edit: In light of dollaaron's correction below, this is much less significant than I thought at first glance. This part "Scholarship or grant funds will be provided to cover these costs in lieu of a parental contribution" is still above and beyond what many schools do, but is similar to Princeton and Harvard, as he points out.