Not that wealth disparities aren't a serious issue; the issues mentioned for poor parents raising children are certainly real. They just don't strike me as the reason Americans don't parent in a laissez-faire manner. The reasons are more to do with lack of community engagement, lack of trust, misplaced fear of harm, an endless optimizing mindset, and so on. Wealth disparity doesn't even register.
We might say that low income and a lack of social safety net results in increased anxiety about oneself and society, which results in lack of trust, increased fear, etc, and consequently more controlling parenting. This seems plausible enough. But it doesn't well explain the recent massive increase in hypermanaging/helicopter parenting. American society hasn't become more precarious or more dangerous. While income anxiety may be a contributing factor, it probably isn't the proximal cause. There was a phase change in the average parenting ideology in the last 30-40 years that isn't well explained by the usual societal bugbears.