Other than in extreme cases of abuse or neglect, almost all of the long-term impact that parents have on their children’s long term life outcomes (happiness, income, intelligence, health, etc), can be explained via genetics rather than parenting style.
The research is still very unclear, but the study of epigenetics at least suggests that there is a more complex interaction between genetics and environmental factors than a traditional understanding might suggest. See for example the section on epigenetic psychology and psychiatry on Wikipedia: "subjects who carried one or two copies of the short allele of the serotonin transporter promoter polymorphism exhibited higher rates of adult depression and suicidality when exposed to childhood maltreatment when compared to long allele homozygotes with equal ELS exposure".
Regardless of epigenetics though, it also seems crazy to me to assume that family wealth and the context of privilege in which someone is raised has little or no impact — if only in terms of helping to define someone's expectations about what they "deserve" in life, and giving them a cushion to fall back on when taking any sort of financial risk. Unless you think that parental wealth itself is a result of genetics. :)
Fair enough. I guess it depends what you count as abuse or "extreme neglect" — just looking at myself and the people I know best, it seems like we were all influenced by our parents' treatment in key ways, both positive and negative, and I suspect that "a relatively broad normal range" still includes a lot of questionable stuff! (That is, we're all terrible at parenting in one way or another, but probably some of us more than others.)
Obviously in terms of how similar we end up to our parents, that could be highly genetic, but I'm thinking more about the way many people either directly rebel against their parents or who more generally try to compensate for things they feel like their parents did poorly.