Parental income and education are both heritable for the same reason height and hair colour are heritable. Grandparent is directionally correct although the effect of parental education on child education is almost entirely mediated by IQ and Conscientiousness. So people who were middle and upper class in their home country but come to richer countries have children who achieve similar success in their new country even if they end up owning a dry cleaners because they can’t really speak English.

> The heritability of conscientiousness facets and their relationship to IQ and academic achievement

https://www.pnas.org/content/111/42/15273.long

> Genetic research has shown that intelligence makes a major contribution to the heritability of educational achievement. However, we show that other broad domains of behavior such as personality and psychopathology also account for genetic influence on GCSE scores beyond that predicted by intelligence. Together with intelligence, these domains account for 75% of the heritability of GCSE scores. > The high heritability of educational achievement reflects many genetically influenced traits, not just intelligence

https://genepi.qimr.edu.au/contents/p/staff/CV448.pdf

> Our findings confirmed positive associations between IQ and the facets of Competence and Dutifulness (ranging 0.11–0.27), with academic achievement showing correlations of 0.27 and 0.15 with these same facets and 0.15 with Deliberation. All conscientiousness facets were influenced by genes (broad sense heritabilities ranging 0.18–0.49) and unique environment, but common environment was judged unimportant.