The thought process goes like this:
Democratic Party = Donkey = Jackass ---> Jackass Party
"Jackass Party" has a two- fold meaning. On one level it refers to the Democratic Party's mascot- the Donkey. On another level it expresses the common Republican view that Democrats are Jackasses - hence the "Jackass Party" (or the party of jackasses who's mascot is a jackass).
GUY ON THE STREET: Can you believe what Obama just did?! What a bad idea!
REPUBLICAN: That's what happens when you have a member of the Jackass Party running things.
DEMOCRAT: I love ObamaCare, and marijuana, and political correctness, and Hillary Clinton, and Planned Parenthood, and welfare, and government programs.
REPUBLICAN: So. You're a member of the Jackass Party, huh?