I have an additional problem with the studies. How do you determine how parenting style influenced the child? You cannot re-run the experiment. A correlation between the scores of parents and adopted children (in any test) implies an influence. However, it is not required.

To give a simple example, take some test which has just two options: "Yes" and "No", both for parents and children. Assume that 50% of the parents have "yes" and 50 "no". There are two different parenting styles: One guarantees that the children will get "yes", and one will guarantee "no". Clearly this is a large impact of parenting! But what happens if all parents choose between those two styles with 50% probability? You won't see any correlation between the parents and the children. While this example is unrealistic, it is not so absurd to imagine some children which try to become different from their parents, while others try to become similar. The parents are important in that case, but a simple correlation will not show this.

Twin studies are better to find influences of parenting.