I believe all parents have a huge responsibility to their children, no matter the age. For a child to be dead that many hours and have no one call for help is outrageous to me. Even if there was nothing that could be done for the teen, someone should have called authorities immediately.
Any parent that lets their child suffer and reach such a weight to cause death should face death themselves. In fact, they should likewise be starved to the extent that they assume the exact same weight. This is a classic case of an eye for an eye and a tooth for a tooth.
The parents should not be killed. The parents should go to jail for a 5 years. If the parents suffer death then we will lose 2 more people than we should have to loose. If the parents go to prison then they will quickly learn what they have done wrong. Also if the parents have more kids the kids should be put up for adoption.
I think that another punishment, rather than the death penalty, would be appropriate for the parents who did not call emergency rescue services for their dying or dead child. They need to be examined for mental illness and treated if possible. If they acted out of cruelty and malice, then life in prison is the punishment I would hope they receive.
No sane parent allows a teen to starve to death, then waits a day and half before calling 911. There were some medical issues involved, but 37 pounds is way too underweight for a normal-sized teen. There had to be some kind of insanity involved, whether the parents caused the death or merely stood aside while the teen starved. The civilized world does not put mentally disturbed people to death.