Nast's cartoon poked fun at the New York Herald, who had been critical of rumors that President Ulysses Grant would seek a third term. The cartoon featured various interest groups represented by animals with a donkey in a lion's skin labeled "Caesarism" at the center. At the edge of a pit was an elephant labeled "The Republican Vote," who Nast portrayed as fearing the Democrat's scare tactic.
Nast would continue to use the animals in cartoons as shorthand for the parties. By the 1880s others were using it as a symbol of the party as well.
Like with Jackson and the jackass, Republicans grabbed hold of the elephant as a symbol of strength and it too stuck. Now both parties have animals as symbols and both symbols came about not through their own selection, but because it was more or less selected for them. The Republicans have even adopted the elephant as an official symbol of the party. The Democrats, however, have left their poor ass hanging in the wind and have never adopted it as an official symbol.