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It has been COMMON in the past 20 years and longer for hospitals and nursing homes to stop nutrition and fluids by bunting to  private and "in-facility" hospice that provide narcotic pain medicine only to facilitate transport to that "great good night". Even if the patient is deigned terminal none of these things should be going on. Even non- terminal patients not in hospice that may be considered difficult to manage or severely incapacitated (Alzheimer's) are having fluids removed with permission of their families- that being their only source of hydration.
 Palliative care is a whole new medical area of "expertise" where the end goal is always a painless (to the observer) death. Physicians specialize in it.  Many of these organizations are NO "catholic" and provide many services to the family such as respite care and psychological counseling which manipulate families in their favor.
If a hospice patient gets pneumonia, which can be treated successfully even as a comfort measure for a terminal patient, they will be summarily kicked out of the hospice program with it's special funding because it is considered a life- affirming treatment (not allowed!), and they are in the business of terminal care only. Of course, the patient can re-apply to hospice when they decide they will forgo actual treatment and die within 6 months.
Most staff feel that they are doing "God's work" by working with the diagnosed "dying" but the line between compassion and euthanizing is easily blurred when your goal is not to sustain life.
I was supporting a friend whose husband was critically ill in ICU  five or six years ago, on ECMO but still conscious and coherent. They kept asking him if  he wanted to stop treatment and his answer was "I want to live!" and "what are the other options?" A palliative care doctor started pressuring his wife ( my friend) to stop treatment and with her permission I answered for her. I told her the patient was conscious and that they should listen to him, and also that "we are Catholic" and must favor sustaining life, especially if it is the will of the patient. The doctor, working for a Catholic agency was highly insulted. Yes, he shortly died thereafter, but it was not an accelerated death.
Now ECMO is not an ordinary measure to prolong life but it was the patient's choice to make- they didn't want to hear that.
A few times I have been asked to remove an IV that was sustaining a patient and had to refuse to the detriment of my employment.  I was also asked to give what I considered a lethal dose of morphine to a frail old lady that blew out her ventricle from a major heart attack ( yes she was going to die) but I couldn't speed up her death with that dose. The medical environment it full of pitfalls for serious Catholics that know their faith. Most Catholics, including "religious" always put the false "medical religion" before their Catholic faith. Medicine-ism " Tɾυmρs Catholicism every time.

I shared the gist of your comments with my parents this evening, and we are adamant that my father is not going to a nursing home, and he is not going to hospice.

When my father last saw his doctor, in October (further medical care is pointless), they were supposed to have hospice to call us, but alas, in this part of the country, there is this quaint little practice of "people not calling you back when they say they are going to".  I don't know if it's just irresponsibility, the laid-back Southern lifestyle that prevails in many quarters here, or a dose of "screw those [pseudo-] Yankee transplants*".  But moral of the story, hospice never called, we finally called the doctor's office to see what was up, and in the meantime, decided not to go that route.  After reading your comments, I'm confident we made the right decision.  If we need pain medicine down the road, after we've run out, and if my father is still living at that point (if you'd told me that he would live to celebrate Father's Day, I'd have said you were crazy), then we shall just have to cross that bridge when we come to it.  His doctor, God bless him, is willing to make house calls, but my father wishes no medical care, just wants to die at home with his family.

* - whether we are "Yankees" or Southerners is a matter of opinion.  The part of the country we come from, is a little bit of both, kind of like if the Midwest and the South had a baby.  We practiced the "midwestern goodbye", i.e., when a guest gets ready to leave after a visit, you spend 45 minutes saying goodbye, in a slow procession from the living room to the driveway --- a ritual that always drove me bonkers growing up! --- and we thought grits were some weird food that other people eat.  Hard to say.

* - whether we are "Yankees" or Southerners is a matter of opinion.  The part of the country we come from, is a little bit of both, kind of like if the Midwest and the South had a baby.  We practiced the "midwestern goodbye", i.e., when a guest gets ready to leave after a visit, you spend 45 minutes saying goodbye, in a slow procession from the living room to the driveway --- a ritual that always drove me bonkers growing up! --- and we thought grits were some weird food that other people eat.  Hard to say.

This description hits me as familiar -- I think we used to do this, I'm afraid. I'm from the Midwest myself.