I believe the General Social Survey (GSS) has this. And there are the longitudinal education surveys that have been launched every few years for decades: Baccalaureate & Beyond, Beginning PostSecondary, National Educational Longitudinal Survey, etc.
[edit: added links]
I've gotten used to blogs and websites that reject comments that contain too many links so I no longer worry about providing links. But I've been requested to supply links so here's a decent one for the educational longitudinal surveys, at the National Center for Education Statistics. There are a number of additional surveys that all have acronyms such as NLS, NELS, or ELS that for some reason are not listed but this is a very good starting point. It also does not link to the GSS.
And here's another NCES webpage that gives an overall introduction to some of the longitudinal surveys -- but only four of them.