For one to end up being in that state would be a direct result of accumulated actions/karma from previous deeds. For the family members, having to go through the turmoil, the pain, the sadness and emotional rollercoaster is also a direct result of previous deeds. Therefore both parties have to undergo and experience it entirely to exhaust the karma that has been created from past actions.
If the parent asked to be euthanised, it would mean they choose to end their "suffering" early, and in the next life, would have to suffer again, this time maybe heavier. If the kid agrees, it may be for a few reasons:
1. to end the pain of their parent
2. to put their parent and others out of misery
3. although difficult to do, they may want to choose the best and most beneficial solution for both parties including minimizing the burden that the kid(s) would have to deal with should they not opt to euthanise the parent.
Therefore, choosing euthanasia is ending suffering and what you otherwise might have to endure, early. If we believe in karma, we would know that no matter what, it will return if not fully exhausted. And everyday we accumulate more just by living. If we choose to euthanise our parent, even at their consent, the result willl come back and bite us twice as hard. I wouldn't put my kids through that. I'd rather my kids say mantras by my bedside..at least my consciousness will have some imprints.