This is complete crap. My parents both left school at 15 with no qualifications. I have an honours degree and my brother has a phd. My father was a smart man and got some vocational qualifications but as a miners son in the 60's university seemed out of reach to him.
In the 1960s university takeup was around 6% (depending on if we're talking pre- or post-Robbins: it was higher by the end of the decade). Today it's around 35%. There are today far, far, fewer people who could or should have gone to university but didn't. As a result of that, a whole slew of educational opportunities have disappeared (WEA, for example).
It's not a straightforward comparison, because a lot of non-university qualifications then are university qualifications now. For example, there are now no Cert Eds, but the distinction between your mother having done a two year residential teacher training course or a three year residential first degree may not be useful when considering parental education, and the same goes a fortiori for nursing qualifications. But in general terms university educations are far, far more common and far, far more accessible than they were a generation or two ago.
This isn't a debate about whether this is right, wrong, wise or unwise. As a simple statement of fact, at least five times more parents of children today will have degrees than was the case when I was at school (I started school in the late 1960s).
The disparity won't be as great if you ask the question "what proportion of children have at least one graduate parent?" because graduates tend to intermarry, .
But the massive increase in university take up begins with the cohorts born in the early 1960s (who now might have children at university themselves) and really starts to ramp up with cohort born in the mid 1970s through to the early 1980s (who probably form a large proportion of current school parents). The disparity in educational attainment between the people stood outside the school gate in the leafy suburbs of London and outside the school gate in a rustbelt town in the North East is massively, massively greater than it was a generation ago.