Bullfinch is a name given to two groups of passerine birds. Bullfinch is a name given to two groups of passerine birds. The true bullfinches are thick-billed finches in the passerine family Fringillidae. They comprise the genus Pyrrhula. These birds are restricted to the Old World, and they include the following: There is an unrelated group of New World birds also named as bullfinches because of their superficial resemblances to the above species. These passerines were placed in the large bunting and American 'sparrow' family Emberizidae, but are now considered tanagers (Thraupidae). Four are in the genus Loxigilla: