To unlock all the theme park mascots, all you have to do is play one game of each of the games included in Family Game Night 3. Once you have, this achievement will unlock. If you need a check list, here is a list of all the mascots.
“Twister” Dancer Mascot
“Mouse Trap” Mouse Mascot
“Yahtzee Hands Down” Card Shark Mascot
“Clue” Policeman Mascot
“The Game of Life” Billionaire Mascot