I am an attorney and I handle probate matters. Do not take this as creating an attorney-client relationship. It is my opinion about general law.
DrasticDreamer and ninyjobs are basically right.
If the spouse left behind a Will, then the Will controls. If there are conflicting Wills, the court will decide which one controls.
If there is no Will then the person died “intestate”. In that situation the surviving spouse would ordinarily be appointed the administrator of the estate, but there are some exceptions. As just one example, if the surviving spouse refused to open a probate case on the estate or delayed unreasonably in doing so, I can see where the parents could file a probate on their own and one or even possibly both of them could be appointed as administrator. That appointment might or might not be set aside if the surviving spouse challenged it. Basically, anyone can file a probate case. The court has a lot of latitude to adjust things like who stays in charge as administrator after the case is opened.