I wonder whether it is the father's educational achievements per se, or rather their attitude towards education, that is the best predictor.
It is interesting reading everyone's personal examples of families that buck the trend; I guess I do too- parents have 5 O-levels each, then both went on to get catering qualifications and trained as chefs. They now both work in bakeries- one more a family business, the other a factory. Academic qualifications were never a focus of theirs, whilst I've just completed my MSc at Oxford and am about to start my PhD here.
They often say they "don't know where it comes from" (with regards to my love of academia)- so what about their attitudes towards education? Well, neither of them ever really pushed me or encouraged academic success specifically- in fact, they're both pretty laid back and wouldn't mind if I worked in retail for the rest of my life (as long as I was happy). There's no history of academic success on either side of my family, either; in fact, one of my grandfather's can't read or write...
Sometimes it is self-motivated. However, there will always be outliers and I can see how the father's educational achievements could have an impact on their child, but I do think it's more about attitude and the environment that the child is brought up in, rather than the educational achievements per se.