Not really, but no...
Whether parents like it or not, approve or it or not, or wish they had some control over the situation or not...
Parents shouldn't be allowed to have a shred of say beyond sharing their own thoughts, opinions, etc. with their daughter about her situation. It's her body. It's her baby. It's her mistake. It will be her consequences to pay, and even if parents wish they could prevent her from having to pay whatever consequences they see they still shouldn't have any say in the matter, other than whether they're willing/able to support the daughter (financially, emotionally), and whether they're willing/able to order guidance and help with caring for the baby if she has it and if she keeps it.
I'm an adult woman, and I've been one for about forty years. Even with that, I continue to have to deal with other adults (my parents aren't alive any longer, but I deal with others who "appoint themselves" parents of other people, especially women, adult or not). I cannot put into words the disgusting and overbearing tendency of a lot of people to want to tell others what to do for whatever reasons they believe they should have some say in the matter.
In a whole lot of cases of teen pregnancy, the reason there's a teen pregnancy at all is because the young woman's parent/parents failed to instill in her the maturity and good sense not to allow herself to have a pregnancy or not to want a pregnancy because she thinks that's her route to being seen as a grown-up or getting out from under her parents' house). In other cases, the girls' parents may have been perfect, but kids don't live in a vacuum; and sometimes they get into situations and make bad/risky judgments. Nobody in this world has had (whether s/he knows it or not) made some bad judgments, so a girl like that shouldn't be treated as less human just because her particular bad judgment resulted in a teen pregnancy.
Teens in that situation need supportive, loving, caring, people around them to encourage them, and guide them, into becoming MORE "human beings in their own right", not less.
As far as the broader costs of teen pregnancies goes, it's an issue that needs to be addressed; and it's one that requires leadership and education (and education of parents of daughters AND sons - not just sex education for kids). The problem is a bigger/deeper one that just a matter of money or even the poor judgment of any individual kid (no matter how many individual kids there are out there).