To be frank: if in scenario (2) the parent is working full time, as (1) why shouldn't their income be approximately the same?
Because the salaries are vastly different. Once you start create such an illogical tax/welfare system where there's no incentive to work harder/take on more responsibilities then people stop doing so.
So in the above scenario, person #1 would probably have a job which involves a lot of responsibility, long hours and probably travel.
Person #2 would probably have a job where you turn up, do what the boss says then goes home.
If there's no real difference, why doesn't #1 quit his job and go be person #2 instead? Yet we need people like #1 to earn more and therefore pay more tax so person #2 can get all his benefits.
We already see this as many people stopped doing overtime because of tax credits. We all know people who've turned jobs down because it's not worth it once you take off benefits.