Section 3.6. "Family background" of the pdf you linked to discusses the impact that parent's educational attainment has on how much money someone earns as a result of going to college, not whether they attend college. It's not used as a control in a way that's relevant to this discussion.
Do you have another study?
Either way, it's a fact that parental income is the best predictor of future income. Not educational attainment.
Do you have another study?
Either way, it's a fact that parental income is the best predictor of future income. Not educational attainment.