Does parental income influence schooling outcomes?

According to many studies using regression analysis that I have seen, family income and socio-economic class matter quite significantly in the determination of schooling outcomes.

It therefore comes as no surprise that as reported by the BBC, Plug & Vijverberg (2005). "Does family income matter for schooling outcomes?: using adoptees as a natural experiment", Economic Journal comes to a similar conclusion. This paper claims to differentiate between parental ability and parental incomes.

I am somewhat uncomfortable about the last paragraph of the BBC article. Reporters sometimes miscommunicate and '(t)he reason is that more than half the transfer of ability from one generation to the next is genetic, as they showed in a previous study' strikes me as a possible simplification. I want to read that article to see how they got this statistic. What does 'transfer of ability' mean? How is ability measured?