You asked what the legal terminology is for, I assume, a case that you could successfully bring against this nursing home facility. The reason that I was asking those questions is that each state has different "causes of action" (a cause of action is a set of elements that, if proven, entitle the plaintiff to damages). So one state might have a certain cause of action, while another one doesn't. Also there is the issue of "standing", meaning that you can only sue for your own damages and those that you "own". For example, suppose you were part of a car accident. Someone hit the car behind you, which caused that car to hit yours. The car behind you suffered extensive damage and someone died. Your car suffered minor damage and no physical injury. While the person being sued could be sued for wrongful death, the party that would bring that suit would be the car behind you, not you (as you didn't "suffer" the wrongful death). "Standing" is where you have to show that you have some "injury in fact" from the behavior.
So it's clear that your father would have had standing, and thus your father's estate gets that same standing. Your mother has some sort of standing for loss of consortium (affection), in that she no longer has your father around. If she's the executor, she can bring the case on behalf of your father (the father's estate "inherits" his case, remember). And the beneficiaries of the estate share in the money recovered in such a lawsuit.
All of this is to point out that for YOU personally to have a case for how your father was treated, you have to show how you have been harmed due to their treatment of your father. Again, it's the harm to you that you'd be suing them for, not the harm to your father. Only the executor or administrator of the estate could bring the case complaining about your father's injuries. Your mother could sue for how she was harmed due to their actions and inactions. And you could sue for how you were harmed. Basically since you can only sue for your harm, and it doesn't appear that you're saying that they did anything TO you or directly took action against you to harm you, but only through their actions or inactions towards your father, your case would be called "loss of parental consortium".
Again, the state where this occurred is important, as between 17-20 states recognize a cause of action for parental consortium. And that means that between 30 and 33 states do NOT recognize that cause of action. Minnesota is one of the majority of states that does not recognize that cause of action:
Lefto v. Hoggsbreath Enterprises, Inc., 567 N.W.2d 746 (Minn. App. 1997), aff’d, 581 N.W.2d 855 (Minn. 1998)
(‘‘[w]e conclude . . . that . . . a new cause of action on behalf of a child for the loss of parental consortium should not be recognized’’)
Salin v. Kloempken, 322 N.W.2d 736 (Minn. 1982)
"...a new cause of action on behalf of a child for the loss of parental consortium should not be recognized."
These cases basically say that you can't sue for the damages to the relationship between you and a parent (and other cases hold that a parent can't sue for damage to the relationship between parent and child if the child is injured/killed). The cases lay out the reasoning, but the main point is that you personally would not have a case that you could sue the nursing home, as Minnesota does not recognize this cause of action.
There IS a case for spousal loss of consortium, and your mother can pursue that on her own. There'd also be a case for personal injury that the estate would have, and the executor / administrator can bring that on behalf of the estate (but not personally, as that case belonged to your father... as he is no longer around it would have to be brought by a legal representative of your father, which in the estate context is an administrator/executor).
Aside from the lawsuits mentioned above, the ONLY other recourse would be to file a complaint with the Minnesota Board of Examiners for Nursing Home Administrators: https://mn.gov/boards/nursing-home/complaints/
I know this is probably not what you wanted to hear, but it is the law. I hope that clears things up anyway. If you have any other questions, please let me know. If not, and you have not yet, please rate my answer AND press the "submit" button, if applicable. Please note that I don't get any credit for the time and effort that I spent on this answer unless and until you rate it positively (3 or more stars). Look for the stars on your screen (★★★★★). Thank you, ***** ***** luck to you!