It definitely contributes to a lack of social mobility. If you are competing with others for zero-sum access to the upper reaches of society and they get substantial assistance from their parents then you don't have as much of a fair chance.

The question becomes - how much of a factor does it play? I would think less than the intangible but non material benefits of having higher income parents (your upbringing, the lessons learned from your parents, their professional connections, a safety net that lets you take risk) but more than I would like to have admitted to myself a decade ago.