In a whimsical moment, the Green Party National Committee has been thinking about a possible official animal that would symbolize the Green Party of the U.S., since the Democratic Party has the donkey and the Republican Party has the elephant. According to Green News and Opinion (here), the choices, in order of support, are (1) Turtle; (2) Frog; (3) not to have an animal symbol; (4) Bee; (5) Sunflower.
Minor parties have not generally been attached to animal symbols, although the Prohibition Party consistently used a camel; the 1912 Progressive Party used a bull moose; and both the Reform and Constitution Parties used an eagle.