Where exactly in the USA this happens all the time? At the rate illegal immigrants have children you should hear about them obtaining a way to stay everyday. Fact is that Anchor Babies are not as successful as some americans would like to think they are. Having a baby in the US gives no rights to stay in the country for the parents until that baby is of age to sponsor and has a minimum income (so the parents can be sponsored). I have known several families that have been deported from the US and they have US born children. They do not deport the children but when the parents go, they children do as well. The cases you hear about are extremely rare and as such they get all of the publicity (as opposed to cases where parents are forced out of the country because no one cares about illegal immigrants). The cases that allow parents to stay are not based on their relationship to the US born baby, but to other issues, from compassionate grounds to abused women.
PS: It is amazing how oblivous Americans are about immigration, until they face emigrating to, for example, Canada - and even then they have no clue what it all entails, many think they will be immediately citizens of Canada, will lose they US citizenship or that just by marrying a Canadian it gives them the right to stay in Canada as they please. Nothing farther from the truth. So, guess what? The process is the same in the US. No one is entitled to stay in the US automatically, everyone that is foreign born has to go through the immigration process in the US too. Canada, however, has immigration classes that allow a person to come and stay in Canada without having a relationship with a Canadian (or Canadian PR). The US lacks those classes and therefore illegal immigration is more broad (as highly skilled people are found to be illegal immigratns as well). The US employment class takes forever and it can be denied even to people with great skills. US citizens often defend their immigration process asking people to go back to their countries to get in-line for immigration while in most cases there is no such line - the line argument is unnecessary but politicians use it well taking advantage of american's ignorance. The US has a few lines, and most include the family class. Other two lines are Entrepeneur and Employment. Canada has many more lines for those not linked by family ties to Canada; many more than the US has, and that is why Canada in the end attracts more highly skilled immigrant every year (thus reducing the need for marriages of convenience).