Turns out that it doesn't matter how poor or uneducated the parents, the students still perform in the top 20%, including maths and science (the latter being particularly suspect given the strong showing of fundamentalist religion in the homeschooling world).
Going back to the reference paper and pulling another reference[2], it clearly states in the abstract that while they found significantly higher rates of performance in the home-schooled kids, they also had a significantly more educated parents and significantly higher family income... and also clearly state that since their experiment wasn't controlled, you shouldn't read too much into it.
These numbers in that link are cooked. They're nonsense. That link is using weasel words, apples-to-oranges comparisons, and pulling info from non-neutral parties. Is home schooling better or worse? Well, it depends; it is better for some, and there are certainly some myths. But that link is cooked.
