I think the Irish Rovers MAY be who you are looking for, though this isn't the Tyrone band I was thinking of. The Canadian-based Irish Rovers did have a big hit with The Unicorn - "There were green alligators and long necked geese, Some humpty-back camels and some chimpanzees, Some cats and rats and elephants, but sure as you're born, The loveliest of all was the Unicorn."
https://theirishroversmusic.com/our-story/