Do you have to own ordinary cows/sheep/chickens for the "special" kind to become available?
It is the first bazaar of year three, I own no animals, and I've reset a half dozen times...
Unlocking 3rd Year Animals
I'm not sure. That might be the case though. At the first bazaar of my third year there were plenty of "special" animals. I already had multiple livestock animals though. The guide doesn't say anything about this. I'd just buy one of each normal animal. That should unlock all special animals if that's the issue.