I think you might be misinterpreting parent. He's just talking etymology, and he has a point. The eu- prefix means good/pleasant/easy, the rest of the word mean

I think you might be misinterpreting parent. He's just talking etymology, and he has a point. The eu- prefix means good/pleasant/easy, the rest of the word means death. So euthanasia, as a category of killing, implies that it's giving a good death, where otherwise there would be a bad death. A synonym would be mercy-killing. So you can argue that if the animal is not yet ill, it can't be euthanasia. Of course you could imagine non-standard scenarios where killing someone healthy might be considered euthanasia, like intentionally inhaling fire (thus rendering self unconscious) when you expect to burn to death. We're just talking semantics, not debating choices.