Does emergency respite care exist?

Speak with one of their local hospital's social workers. Also make sure the police department and EMS in their town have your contact information. The police should be doing regular wellness checks on your parents if they aren't already.
You're right that your shouldn't impose on your friends to do wellness checks on your parents or to handle any emergency situation that may arise. Really though, they most certainly should have hired help coming in seven days a week. Even if it is only for a few hours a day with regular wellness checks from the police too.
There are no emergency in-home respite care service agencies that I ever heard of. Some homecare agencies may be able to accommodate temporary live-in care but may not be able to place a live-in caregiver at a moment's notice.
If your mother falls and gets taken to the hospital, the paramedics or police will also take your father to the hospital if they know in advance that he has dementia and there is no one but your mom in the home to care for him. Make sure they know this.
If your mom has to be admitted into the hospital for a stay, they hospital will place your father in nursing home/memory care facility as respite.

Yes, check with your local Senior Living Communities that offer a continuum of care..independent apartments, Assisted Living and Memory Care. Often they have a beautiful furnished apartment set aside for respite care( think luxury hotel) where the guests are invited to dine in the dining room, go to movie theater and other amenities. Perhaps,you can present it to mom and dad as their own mini vacation...and it also serves the purpose of introducing other future care options to them in a very low key way, while having the security of being safe. The other option is to maybe look into hiring a student from a college nursing program if your folks have an extra bedroom. Many students are able to use such a situation toward their field experience hours. Best of luck and enjoy your well deserved trip!

Sometimes couples go to respite care. It might be a nice vacation for your Mom. Sounds like there are lots of nice places near them.

Jadee3: Respite care exists in your area of Hoboken, NJ. As it can be googled, I will not list it for purposes of no advertising here on the forum.

Visiting Angels or call the Care Advisor here.

Yes , I am in NJ and respite care for any amount of time does exist . Contact local nursing homes and rehabilitation facilities. Each has their own rules . You will be surprised at what you find . I am considering using them once a month for my mom . I am an only child and besides vacations , I too need a much needed break . I am going to try getting an aid this week if that doesn’t work , I am committing to once a month for a week in a facility . I am fried and my health and patience is being severely compromised with 24 hr care I give . I work as well and I value those days away . Please don’t think I am mean … we all need time to regenerate ourselves . Do it today !

You said your dad is on a morphine drip, but not under hospice care. Who is administering this drip? That's who I would ask first about arranging for emergency respite.

Bear in mind, though, it's likely the cost of the respite would be coming out of your or your parents' pocket(s).

I have researched this very scenario.

I was told (where I live) if a caregiver sustains a fall bad/other health crises & needs emergency transportation to ER, without a backup caregiver able to step in, (nearby relative or willing friend/neighbour) the dependant person would need to be transported also - as a 'social admit'. Then a Social Worker would get busy trying to locate NOK or another relative to collect. If none available, the person would be kept in a hospital/rehab bed, even NH bed until other arrangements made.

EMS crew would need to be informed that the other elder could not be left alone.

This plan has a big hole...

What if the caregiving elder could not communicate? Be very unwell/banged their head etc?

Then what if the other could 'showtime' well?

I suppose a sign on the fridge could be made..

I'd try for a friend/neighbour doing a welfare phone call evening & night.

I ended up rationally trying to think how unlikely it was & set up what I could.

Hope you can enjoy your trip 😃

Hospice does cover Dementia. My Husband with a diagnosis of Alzheimer's was on Hospice for almost 3 years.
Does you mom have an "alert" device? Some can be set so that a fall would be detected. If she did not answer Emergency Services would be sent. (some will send if movement has not been detected for a specific period of time)
I have seen the Apple Watch on a commercial and that also detected a fall .
Do mom and dad have friends that can check in on them?
Do you have a friend that would check in? either they could do it as a "favor" or you could pay them to check in. How much do you want to be done? You say you can not impose on friends as dad is a lot to handle. But there is not much to do checking in on them. But if you are doing daily tasks and chores for them that is different.
You could arrange a week stay at a local Assisted Living facility. They would have the watchful eye of staff. You would not have to have someone check in on them. And it would give your mom a bit of a break knowing she does not have to all the caregiving. Many AL will do a Respite stay like this as they would want them to become residents when and if the time came.

If your dad has end-stage dementia (6 months or less), he can use Hospice, which does cover respite care.