That is what I often wonder on the larger discussion of "privlege". What draws the line between your parents or grandparents having worked for a better life for you and having some sort of shameful "privelege"? I understand that there is a difference when it comes down to historical oppressions of entire categories of people, but I feel that many people often confuse that and "rich parents vs. poor parents", seeking to remedy some percieved unjust inequality where the inequality comes from earlier sacrifices leading to more positive outcomes.
We have a saying in Britain, that few people remember now: "rise with your class, not above it".
That comes from an understanding that making the best of living under a system that creates opportunities to be rich and opportunities to be poor is not the same as altering that system.
Really, the idea of privilege as people use it now is too individualistic to make sense at the macro level.