Is there a correlation between GRE scores and family income?

According to this article on the Fair Test website,

The GRE is particularly susceptible to the influence of socioeconomic class. ETS' own research has shown a strong relationship between family background and test scores. One study of applicants who scored between 750 and 800 on the exam found that only 4% of these high-scoring test-takers had fathers who had not completed high school; around half had fathers with bachelor's degrees or more, and of these, a whopping 90% had fathers with graduate or professional degrees. When family income was held constant, most of the test score differences between races disappeared or shrank dramatically.

The correlation between GRE scores and future academic success is much lower, according to the same source.

The ability of the GRE to predict first-year graduate grades is incredibly weak, according to data from the test's manufacturer. In one ETS study of 12,000 test takers, the exam accounted for a mere 9% of the differences (or variation) among students' first-year grades.2 Undergraduate grades proved to be a stronger predictor of academic success, explaining 14% of the variation in graduate school grades. An independent non-ETS study found an even weaker relationship between test scores and academic achievement - just 6% of the variation in grades could be predicted by GRE scores.

Another article from The Atlantic: The problem with GRE

ETS studies have also concluded the GRE particularly underpredicts for women over 25, who represent more than half of female test-takers. Research from as far back as the 1960s leads experts to believe that the inconsistencies in GRE performance trace to a combination of factors including access to coaching, a disparity in educational opportunities that better prepare some students for the test, the content of the test, the way students are tested, and even the student’s own insecurities regarding race and gender. Sternberg puts it bluntly: “The GRE is a proxy for asking ‘Are you rich?’ ‘Are you white?’ ‘Are you male?’”