@GirlWithAGuitar
It's well short of obvious to about half the people on this thread. And keep in mind, we're only hearing OP's side. She says he knew they were hers, but she also seems to think he at some point agreed to living with them, when he clearly doesn't see it that way. I doubt he ever enquired into the proper ownership of the dogs that lived with his in-laws. Most people wouldn't.
Regardless, OP admits that in three years, they never once discussed this. In the eight years before that, she couldn't be bothered to train the dogs not to yap, so that someone other than OP might actually want them. Then, rather than discussing their future with her partner at that time, she just assumes based on nothing that he'll be fine taking them. Meanwhile, the dogs get three years older. Now it's an emergency because her parents can no longer care for them, no one other than OP wants a pair of yappy senior dogs, and there's no plan B because OP couldn't be arsed to think of any of this until now.
So yeah, maybe he's not the greatest person ever for vetoing them. But if the only choices here are "DH agrees to live with the dogs" and "the dogs get PTS because they're unadoptable," back up and ask how things got to that point. They aren't his dogs, and their welfare isn't his responsibility - it's OP's. And if OP gave a flying fuck about the dogs' well-being as living creatures, rather than as accessories to her own life who exist for the purpose of making her happy, she'd never have put them in this situation. She'd have trained them not to yap before dumping them on her parents. She wouldn't have left them in limbo for three years. She'd have discussed this with her DH at the beginning of the relationship, and either ended it early or made other arrangements for the dogs when they were 8 rather than 11.
None of that is DH's fault. All of it is OP's. This problem wouldn't exist if OP were a responsible dog owner, and the fact that she's clearly not is probably part of why her husband doesn't want to deal with her dogs. He probably thinks he'll get stuck taking care of them because she can't be arsed to do it properly. And he's probably right.