As lankeela says, I think you do know the answer.
I know that part of you is also wanting to take the option of giving the dog to your ex. (it will save you making the decision and take everything out of your hands). But is that really what you want?
The vet would not have advised this if he/she did not think it was the only way. Believe me I have not met a vet yet that wanted to put a healthy dog to sleep. It is not a nice thing to do, and not something they become a vet to do!
You owe it to your dog to let him go in the environment that he has known all his life with the people he has known all his life. It would be cruel to rehome a healthy dog of this age, let alone one who is getting confused anyhow. At your house he knows the rules, where the water bowl is, where the food is, where he is allowed to sleep etc. and he is still getting confused what would he feel like in a strange house?
I have had to overule my partner over issues like this - some people just cannot let go (I don't blame them - they just cannot do it). I choose to be with my dogs at the time, but my partner chooses not to be there, people are different.
I think you know that the decision is with you, you have to be strong and do what is in your heart and head. Believe me if you do not and allow the parents to have the dog (and he ends up in a worse state if/when you get him back) you will never forgive yourself. The end will come one day and if the decision is not made now you will have this problem again one day soon.
You know in your heart that you have looked after the dog well, don't let your ex put doubts into your mind - he is hitting you where he knows it hurts.
You are the one who has to live with this, don't let these few weeks spoil the many happy memories your dog has given you.
Take care - believe in yourself (you do know best) and be strong!
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