Short for unisex party. Meaning that the party includes all genders.
"Hey, are you coming to her uni party later?"
"I was thinking for my birthday, I want to have a uni party."
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Uni-party is a term used to describe politicians who are funded by corporations and/or lobbyists as a whole, as if to create a whole party dedicated to said politicians
"The Uni-party needs to be dissolved in America!"