Cognitive scientist Daniel Willingham outlines the research on poverty and education in the Spring issue of American Educator: "Why does Family Wealth Affect Learning?" When scientists speak of poverty they use the term Socioeconomic Status (SES) to capture not only income, but education level and occupation.
Willingham focuses on two main categories: family investment models and stress models. Family investment theories explore all the ways that higher SES parents tend to invest in their children's social, academic and career-path development. They have more money, for sure, and can afford to live in healthier conditions with better nutrition and higher quality daycare and preschool. But they also spend more time with their children, on average, and engage them in stimulating conversations about the world around them. The stress theories are more directly biological: the high chronic stress common in low SES households floods young developing brains in hormones such as cortisol that directly impacts brain development, particularly in memory formation and regulation of emotions. Willingham points out that these theories of parental investment and stress are not mutually exclusive.
For example, Willingham cites research showing that foster children who had experienced chronic stress responded well if the parents were trained to better recognize signs of distress and responded in a sensitive way. Willingham offers the study as an example about "the buffering effect of warm parenting."
I've written about the SPARK program here in the Akron-Canton area, which is aimed primarily at parents to help them become their children's first teacher. And Helen Neville a neuroscientist at the University of Oregon has been working with low SES kids and their parents on a program to improve attention. Again, the intervention that showed the best results for kids was the one primarily directed at parents.