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.
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.