Sorry, haven't read the full thread, but most of the evidence I have seen suggests that the reason poor people are fatter is because they are more likely to eat energy rich but nutrient poor foods.
You see, the problem with poor people is that they are, well, poor. They have less money than non-poor people to spend on food. Energy rich/nutrient poor is generally cheaper to buy, quicker to prepare (remember, most poor people work), easier to store and cheaper to cook.
Some poor people are bloody fantastic parents, and they manage to tease out their time and budget to produce lovely, cheap and nutritious meals for their DC. Other poor parents are a bit shit, and quite possibly feed their kids Tesco Value biscuits for supper. However, most poor parents sit somewhere in the middle.
Rich parents can also be fantastic, a bit shit, or somewhere in the middle. However, when rich parents are a bit shit, their kids are more likely to end up with half decent food.
It seems to me that a lot of time and money is poured into trying to make every poor parent a fantastic poor parent, since shit rich parents are less of a problem.
The real problem, of course, is that poor people are poor. But it's very important that no one actually says that...