For some, esp. those who choose to be childless, a relationship like that is probably the closest we'll come.
I was happy the startup I worked at during that time allowed me to take a week off as sick pay... sent flowers with a handwritten note from our HR leader the following morning, but I opted to come back after a couple days as I needed to take my mind off things.
I take it you've never lost a child? Because I've lost both and sad as it was, losing a dog doesn't even come close. Losing the dog was sad, but I got over it and eventually adopted another pup. Losing the child was so unimaginably awful that I struggle to find the words...
Police knocking on my door at 2am to tell me. Calling my wife at 2:30am to tell her (she was away on business). Waiting for her to find a flight home. The funeral. Dealing with the estate. Waking up every few weeks feeling like it was all a nightmare, only to realize it was not.
It's fucking awful.
Again, it doesn't exist because it makes no sense.