From what you have told us so far, the tree is your parents' tree. It fell part way in a storm, and now is in danger of falling (the rest of the way) onto their neighbor's property and causing injury. There is no way to presently look at the tree and not see it as dangerous and likely dying.
If your parents choose to let their tree stand (for lack of a better word) rather than removing it, depending upon exactly what happens, they could be liable for an injury to person or property that results. The rest depends upon the full facts, including the nature of the area, the manner in which people use the area, and the facts of any injury.

