It's not obvious that it does. Sure, some of these things correlate with wealth disparities. But there's no reason to think the causal arrow flows from wealth disparity to distrust, lack of community engagement, etc. For example, most communities are wealth-segregated. So within community will not see a vast wealth disparity. Any proposed explanation must also contend with the fact that parenting was to a much larger degree laissez-faire just a couple of generations ago. (Average/median) Wealth disparities haven't grown that much in this time (meaning discounting the culturally irrelevant growth of a handful of tech billionaires).
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.