I don't agree that poverty in itself makes it harder for kids at school, it may be a factor in addition to a lack of parental input.
The poorest people can access the internet even just at a library, they can encourage reading even if books are second hand, or library.
They can teach their own dc through free resources from the internet.
You don't have to be rich to take dc to free museums and galleries.
You can earn 200k and by the time you have paid for a huge mortgage and the lifestyle that comes from having such a well paid job, have less disposable income than a person on benefits. It's not how much you earn but what you do with it that counts.
The parents may be rich, but one of them or both could be really thick and not see the importance of an education for their dc, as they will inherit their wealth anway.