Some people will choose to be stay-at-home parents. There is nothing wrong with that. That is a career. The household income... is the household income, not the wage earner's, because for a couple with children the family is their life project, and the stay-at-home parent's "service" is "paid for" by the wage earner.
If we really want to compare apples to apples then we have to allocate a fraction (say, half) of the wage earner's income to the stay-at-home parent for the purpose of analysis. Alternatively compare singles' income to singles and childless couples' individuals, and separately analyze married-with-children-and-both-working couples, and then separately analyze married-with-children-and-one-stay-at-home-parent couples.