How long after euthanasian should you introduce a new dog?

My parents has a very old lab. Her hips are giving out and it's time. It really is, she's been holding out for a while, but seeing her in pain is unbearable for my parents. The only problem they face is that they have a 7 year old Jack Russell and we know he's going to miss her a lot. They want to get a new playmate for him but they don't want to bring in a new dog before euthanasing the old one. So how long after should they introduce the new puppy?

They wanted me to bring my new pup and see how the JR would react to it, but I said no because I'm afraid the lab might try and do something to her. Also, I don't think it fair as my pup and I live a few hours away and that won't be his playmate. I told them the next dog they introduce to him should be his new friend.