This is just reverse snobbery, though. Employers care, for starters. Someone who's been to Oxford is going to carry a huge amount of privilege with them through their life.
And your parents' level of education does shape you, doesn't it? My MIL (working class, without much education) was obviously a hugely conscientious, education-focussed parent. I know she felt really strongly about getting her kids to do their homework and felt it was really important they learned to read early. But, by the time DP was in secondary school, no way could she have helped her study or given advice about university. By contrast my mum taught me one of my A Level subjects, because she figured she'd do it better than my teachers. And these things snowball and have a huge effect on you - I know my DP has had to work so much harder to get to where she is than I did. You can't not be aware of things like that (unless you are a very selfish person who doesn't consider others).