Seeing the two Yoshis being reunited at the end of the Yoshi board in 1.
Everyone celebrating the player's birthday in 4.
At the end of Story Mode in 4, it's revealed that Bowser's goal was to get in on celebrating the player's birthday like everyone else. Aww. Not that he'll admit it, though.
At the end of Mario Party 5, Bowser's wish was for a strong opponent, meaning that even when you defeated him, he got what he wanted.
The cutscene after the third story mode board has Koopa Kid state that his dream is to be near Bowser in a way that sounds very much like a small child enjoying time with their father.
The 4-player minigame Flower Shower takes place during a Koopa Troopa wedding. Awww...
As said in the Fridge Brilliance page for the series: And in Mario Party 5, notice that DK has his own space and this is the only game where he sometimes stops Bowser when someone lands on a Bowser space with a punch. And in said game, it takes place in the land of dreams and wishes, so DK's wish is to help people.
The ending cutscene of 3. The real Millennium Star allows your character to become the greatest Superstar in the universe, with the background music having that Earn Your Happy Ending feeling to it. This, combined with the credits and their music, also drives home the point that this is the last non-third-party game for the N64.
The end of the mini-game "Shy Guy Shuffle" in Mario Party 10 where the players all sit around a table eating donuts. Also counts as Crowning Moment of Funny as it is one of the few mini-games with post-game animation.
Bowser's interactions in Mario Party: Island Tour's Bowser Tower when you play as Bowser Jr. just scream heartwarming. "Aw, who am I kidding? There's a special spot on my black heart for you, son." Too bad he still bashes you out if you win the mode.
Oh, speaking of Bowser and his son, at the end of Mario Party DS, the heroes want them to play Triangle Twisters.