Identical twins, whether raised together or apart, turn out to be very similar, but one still finds differences in IQ and personality. The cause of those differences must be the different environments experienced by the twins, but can't be characterized by simple variables of the sort listed above: it is not the case that the twin raised by the higher SES family has, on average, the higher IQ! In fact, twins raised in the same family are about as similar as those raised apart, so family shared environment does not produce a measurable influence. See below for a plausible model that accounts for such outcomes.Steve Hsu's explanation involves learning styles, despite the fact that those don't seem highly regarded in the psychology community. One more contradiction, I guess.
By now these results are well understood and accepted by experts, but not by the general population or even policy makers. (See the work of Judith Rich Harris for popular exposition). The naive and still widely held expectation is that, e.g., high SES causes a good learning environment, leading to positive outcomes for children raised in such environments. However, the data suggests that what is really being passed on to the children is the genes of the parent, which are mainly responsible for, e.g., above average IQ outcomes in high SES homes (surprise! high SES parents actually have better genes, on average). Little or no positive effect can be traced to the SES variable for adopted children.
The implications are quite shocking, especially for two groups: high investment parents (because the ability of parents to influence their child's development appears limited) and egalitarians (because the importance of genes and the difficulty in controlling environmental effects seems to support the Social Darwinist position widely held in the previous century).
See here for a bit more on nature, nurture and randomness - calling it "randomness" makes more sense to me then "non-shared environment".
So, does that mean that whatever the parenting style, the effect is negligible? On intelligence and personality at least, maybe parents can have very little influence on the nurture bit. That is surprising.