Generally, the photographer owns the copyright on any photos he or she takes. If there's a person in the photo, you might need to get a release from that person for specific uses. But AFAIK
there are no such regulations about other people's property (i.e. Horses)
Horse Forum can make a rule against posting photos of horses that aren't yours, but that would have a large impact on how useful threads could be. Pretty much all of the photos that people post as color examples are posted by people other than the photographers or the owners of the horses.
Whether the owner has given permission to post pictures of their horses or would get upset about it is between the owner and the person who took the photos.
Not sure why Breezy's threads are being singled out for this?