Poverty is a constant reality in the Peck family. Sure, Rob's age in some ways allows him to escape the drudgery it imposes on his father. But the lack of money to buy things that other people take for granted and the need to stretch resources as much as possible comes up over and over in A Day No Pigs Would Die. At the same time, though, as Papa insists, money doesn't totally define who the Pecks are or what they have. Poverty, while a real burden for the family, can't take away some of the things they really value.
The Peck family is rich in the things that matter; their financial limitations ultimately don't affect them that much.
Papa's assertion that the Peck family is not poor is just a way of making himself feel better about the miserable conditions that he and his family live in.