For many non-citizens who either cross the southern U.S. border illegally, or overstay a visa, giving birth to children is a natural next step in their family's life. A baby who is born in the U.S. becomes a U.S. citizen automatically. But does that mean that the parents receive any added rights to immigrate to the U.S., or at least to avoid removal (deportation)?
The answer is not a simple one.
It's true that U.S. citizens can petition for their parents to receive green cards; but only after the U.S. citizen child turns 21. Staying in the U.S. without permission for all that time could be difficult.
Also, anyone who entered the U.S. illegally (without inspection) faces further complications. One is that, after turning 21 and petitioning for the parents, the child will need to be earning enough money to serve as the parents' financial sponsor (or the family will need to have enough assets to make up the difference).
Another issue is that, unless the parents are still in the U.S. at the end of the 21 years, they will need to go to a visa interview at a U.S. consulate. The consular officer will inquire as to whether each parent is "inadmissible" on various grounds; in particular, for having spent more than 180 days (six months) in the U.S. illegally while over the age of 18. If so, the person might be barred from the U.S. for either three or ten years. See How soon can the U.S.-born child of an undocumented immigrant petition for the parent? for additional discussion of this issue.
Some people can apply for a waiver of the unlawful presence ground of inadmissibility, but only if they can prove that being denied the visa would cause extreme hardship to their own U.S. citizen or lawful permanent resident parents or spouse or U.S. citizen fiance. Hardship to U.S. citizen children does NOT count in this context.
Another thing that your friends might have heard of is a U.S. government policy known as “prosecutorial discretion” or "deferred action." Because U.S. immigration enforcement authorities (“ICE”) cannot deport everyone, they typically concentrate their efforts on high priority cases such as criminals, and drop the cases of people with many ties to the United States, including family.
By granting "prosecutorial discretion," the authorities don't give the parents a green card or anything like it, but they do promise not to deport them for the present.
Under the Trump Administration, ICE has dropped this set of priorities, and attempted to deport virtually any undocumented immigrant with whom it comes into contact. However, among the Biden Harris administration's first acts was to reestablish enforcement priorities and bring back the possibility of deferred action.