"There is nothing that says the parents can automatically stay or that we have to provide benefits."
Correct. We could take a hard line stand and say that citizen-children of illegals will be adopted by law-abiding US citizens while the illegals are deported. That could be too nasty-sounding though. It takes careful thought.
The driving force of immigration, the root of the movement, is the war against terror. Perhaps we could win on that, but we need to think carefully. A baby is not often a terror weapon. An Islamic mother, perhaps. But a Mexican mother who just gave birth and looking for the citizen-lotto? Sleazy, true. Worthy of debate. But not a high terror risk.
Better to keep them out to begin with, obviously.