Sadly, I think the only possible explanation is abuse.
If a mother wanted to give birth (rather than have an abortion) but didn't want to keep the baby, she could just go to a hospital and ask for the baby to be adopted. This is the way to safely give up your baby. Normally people try to hide from social services if they DON'T want their child removed, not if they do. Giving your baby up for adoption is perfectly legal. A mother who was homeless, or an addict could do this, and wouldn't be in trouble for it. Hospitals don't report addicts to the police just for having an addiction.
The only reasons to repeatedly abandon your babies and deny yourself healthcare are that you don't want to become known to the authorities or have no choice. Such as if you are in an abusive situation and fear your abuser. Or there is some other serious crime you are trying to hide.
Although I can understand that people who are illegally in the country might want to stay under the radar, I just don't think that someone would end up pregnant three times if immigration status was their only issue, and they had active choice over their other aspects of their life.
I'm sure that when the parents are found there will be a horrible story behind this.