Not entirely the same, but I brought Cadbury home to four dogs (ages 1, 3, 12, 14 years old). I did dog test him before bringing him home from the shelter and he loves dogs! My dogs also live with two cats so they're pretty used to smaller animals around. They're also mixed breeds with very little to no hunting breeds in them and very low prey drive. I've met a few Barbet at dog competitions and I believe they're bird dogs, correct? Not as challenging as, say, a beagle and a rabbit, but I would definitely be super aware of their interactions.
The first month Cadbury was with us, he lived in a large dog crate. The first couple of days the dogs were interested but within a week he just became part of their regular daily existence.
Once he started using his litterbox consistently, I added an exercise pen to the crate. He divides his time between the crate and the pen. The dogs totally ignored the fact that he is mostly out in the middle of the room with just the pen between them. And he was totally unconcerned with the dogs running past the pen. It got to the point where Cadbury would be lying on one side of the pen with a dog lying right next to him on the other side of the pen.
After about six months, I would let him have run of the whole house while the older two dogs were loose and the younger two were in crates. The older dogs pretty much ignored him and the younger two would just sit and watch him hop around. So one day I let him out of his pen when all of the dogs were loose. I didn't make a big production of it, just opened the pen and walked away. Dogs didn't even realize he was loose for probably half an hour. When he hopped up to them, they just kind of glanced at him and didn't think anything of his presence.
I now let him out in the evenings when I'm home to supervise, and after the dogs have had plenty of exercise so they're more apt to just snooze and ignore him. The older three dogs don't even acknowledge him. The one year old is the only wildcard. She still plays really rough and will chase Cadbury if he starts running around too much. When she gets too excited around him, I attach her leash to the leg of the couch. That way she gets about six feet of freedom but can't chase Cadbury. He can come up and visit with her but hop away if she gets too wound up.
All in all, it took a little over six months to get to this point. However, the dogs are with Cadbury all day every day, I don't know how it would work if your parent's dog only sees your rabbit on occasion.
Good luck!
