This is Samuel and I am sorry to hear of your loss and your family is now enduring this situation.
Here is the Nebraska law which allows your parents, the grandparents to petition the court for Grandparent Visitation rights.
It states the following:
43-1802. Visitation; conditions; order; modification.
(1) A grandparent may seek visitation with his or her minor grandchild if:
(a) The child's parent or parents are deceased;
(b) The marriage of the child's parents has been dissolved or petition for the dissolution of such marriage has been filed, is still pending, but no decree has been entered; or
(c) The parents of the minor child have never been married but paternity has been legally established.
(2) In determining whether a grandparent shall be granted visitation, the court shall require evidence concerning the beneficial nature of the relationship of the grandparent to the child. The evidence may be presented by affidavit and shall demonstrate that a significant beneficial relationship exists, or has existed in the past, between the grandparent and the child and that it would be in the best interests of the child to allow such relationship to continue. Reasonable rights of visitation may be granted when the court determines by clear and convincing evidence that there is, or has been, a significant beneficial relationship between the grandparent and the child, that it is in the best interests of the child that such relationship continue, and that such visitation will not adversely interfere with the parent-child relationship.
(3) The court may modify an order granting or denying such visitation upon a showing that there has been a material change in circumstances which justifies such modification and that the modification would serve the best interests of the child.
And so I suggest they can go to the Clerk of the Court and ask for the Petition to file for Visitation - once that is done, they can serve the fiance.