From this logic has sprung a relentless thirty-year effort to raise welfare benefits, expand welfare eligibility, create new welfare programs, and increase welfare spending. The current proposal from Congress to slow down the automatic growth of welfare spending violates these cardinal tenets of the liberal welfare system and thus has led to cries of alarm from the welfare establishment.

Does this mean that all parents should get on welfare to benefit their children?

I do not think that the article is saying that all parents should get on welfare. It is trying to say that the parents in poverty should at least conside applying for welfare, because it would benefit their children. Applying for welfare would help the children succeed in life because they will be living on a better income. They will also want to do better than their parents did so the will work harder.