Based on what I have read so far in the press, including the police statement, not only has there been an ongoing investigation (preliminary cause of death heat, but still awaiting toxicology results for complete and final verdict), an outside investigator appointed. Parents and neighbors have been forthcoming and offered themselves for questioning. These seem to be the typical steps (although for most parents the investigator would be internal).
I don't know that I have a general sense of what charges have been applied to parents of varying status across the board in such cases. I do recall the case of a teacher forgetting that she was the drop-off person on a day in which her morning schedule was atypical due to having to pick something up before school. Certainly the requirements of law (which someone has listed above) will figure into what charges can be filed--including the elusive concept of intent.
Meanwhile, my heart goes out to any family that loses a child, regardless of circumstance. And I remain grateful that it is others than I who have to make decisions in the aftermath.