This story doesn't pass the smell test, imo. Yes, maybe I'm cynical, but the cops are paid to investigate. I hope they do a little bit more than just accept the family's story at face value.
Scarves and hood ties on small children have proved fatal before, it is a known (probably not as well known as it should be) hazard. Children have been injured and died from hanging scarves and hood ties on slides, and other playground equipment. I think it is possible the dog tugged on the scarf in a tug of war type game and tightened it to the point she died from it.
Anything around a child's neck is never a good idea, as adults it is hard for us to imagine all the scenerios that can happen to a child so I am not blaming the parents.
I second that.