Here is the full list of all 16 Puzzle Party achievements.
Beat the final puzzle with one skip or less.
Beat the final puzzle with two or three skips.
Beat the final puzzle with four or more skips.
Beat the game in one of the other modes excluding "story mode."
Beat the game twice in one of the other modes excluding "story mode."
Beat the game three times in one of the other modes excluding "story mode."
Beat the game four times in one of the other modes excluding "story mode."
Find Elon.
Find Naarah.
Find Gideon.
Find Huxley.
Beat Level Twenty Three.
Beat Level Forty Six.
Beat Level Sixty Nine.
Beat Level Ninety Two.