At risk of sounding like a 1920s Eugenicist who's had a Guardian conversion...
The 1944 Education Act made reasonable education available to all. The 11+ system skimmed a generation of bright, aspirational working-class children out of their background and into the middle-class, just in time to attend the enlarging redbricks and then staff the new universities, and to provide the teachers for the enlarging secondary sector of the 1960s. Rising birthrate, endless demand for graduates. There was funding for rising university participation and most of the people going to university in the fifties and sixties had parents who had left school at 12 or earlier.
Of the generation who didn't go to university in the 1950s and 1960s, their children may well have done: there were a lot more places in a lot more universities, and there was still funding. By the 1980s, you could get a CNAA degree with a couple of Ds at A level on a full grant, and there was always the OU. At each point, a proportion of the remaining families who hadn't gone to university saw their children graduate.
Unfortunately, that was all that happened. That there were a lot of people left doing manual labour, being failed by the education system and assumed to be beyond redemption (see, for example, the educational provision on large post-war housing estates, which was mostly secondary modern) was ignored. As manual jobs declined in availability and pay, you were left with people who were the product of three or four generations of educational failure, who had watched their peers becoming affluent but had themselves, for all sorts of reasons, remained where they were.
Unemployment and deskilling bit in the 1980s, so today some primary schools are dealing with people whose parents and grandparents haven't worked fulltime or at all, and for whom education isn't the enabler some see it as (I'm going back, at 45, to do a full-time PhD) but is a dull, pointless interruption to the pathway towards the dole or subsistence working. Blaming the parents is easy, but it's a lot more complex than that.
When I was at a comp in the 1970s, the bright kids went to university, the less successful for jobs at The Austin (ie, MG Rover) or became typists. Today Longbridge is a flat field of rubble and managers do their own typing. That the school I went to fell into special measures is regrettable, but horribly inevitable, when about 60% of the pupils are from houses where no-one is in work. Bad parenting isn't as much of a choice as people would have you believe, and there's an air of "Let them eat cake".