That's not true and we must be careful not to accept an aristocracy (or oligarchy) as inevitable. That the wealthy will inevitably hold and consolidate power is counter to the history of democracy, a specific rejection of aristocracy, and a system which has worked the other way in country after country to great success economically and politically - great news and great hope for humanity in these times. The foundation of the U.S. is that all are created equal, and after that it's a land of opportunity where merit wins out, not inherited wealth and corruption.
If by that you mean that rich have been getting richer relative to others, that's certainly not true. In the U.S. it's been true since the 1980s, but it fluctuates and in most other advanced economies the proportional difference (the disparity) is much less.
If you mean that there is not downward social mobility for the rich, that's also not true. I don't have the numbers, but I've read about research that overall social mobility (up and down) is significantly less than it was before Reagan. There's also the old American family story: Rags to rags in 3 generations. More concretely, the U.S. used to have higher taxes on the high income, and more importantly, a (much higher?) estate tax: Inherited wealth is not merit, but effective aristocracy. That said, I don't know the inter-generational overall trends of wealthy families.
(Or maybe the parent means something else.)
EDIT: Some clarification + the last sentence