When illegal parents with kids enter US and are detained and deported, can they take the kids back with them if they choose?

Of course they can, their kids are nationals of their parent country

Of curse. Only a monster would abandon their children in a foreign country. Many are monsters though.

Absolutely. In fact, it is their responsibility to ensure they have their children's passports from their country of citizenship, and arrange their transportation home.

IF they abandon their child in the US, they are subject to criminal prosecution for child neglect, abuse, abandonment. If they ever return to US, they go to prison. Meanwhile, they lose all rights to their child (so that child cannot in the future sponsor their immigration). Child actually can be handed over to their country's consulate for repatriation with relatives. Then it would be up to child welfare authorities in that country whether child is returned to parents, or handed over to other relatives, or whatever in accordance with that country's laws.