I think you have it backwards. Poor people cannot rely on their parents because their parents are poor too. Every person I know who grew up rich relied on their parents until… well, forever. Privilege is being able to rely on your family.

It's a kind of free insurance: if a bad financial event happens, you're protected against the worst downsides.

If you're from a poor family and break out of poverty, you're still in a worse financial state than someone from a rich family, even holding income and assets constant. You've got to effectively self insure (by taking fewer risks; being more conservative in investments; cutting down on rich people expenses that help with networking) to plan for the worst case scenario. And rich kids don't even realize how much their families' financial status enables and drives their behavior.

And that's not even starting to count financial expenses to take care of aging parents who can't afford to take care of themselves.

I also think that educated and wealthy parents are better able to support their children; from education and financial literacy to real estate and plain money. However, I do think that this support makes the children more independent when they're adults.