[lapcat]

The full article is behind a paywall, but the abstract doesn't seem to say what you're saying. It does say:

1) Kids and parents have differing perceptions of the parenting style: "Birth-order and family-size differences were found in adolescents' achievement and perceptions of parenting style and parental involvement but not in parents' perceptions of parenting." Thus, your statement "the underlying fact that later children get a different parenting style" doesn't actually seem to be a fact.

2) Parenting style, whether it actually differed or not among children, didn't seem to explain the differences in archievement: "these parenting characteristics did not mediate the differences seen in achievement by birth order and family size."