The flying letter leading to World 5 shows Peach being threatened by a Goomba Tower. The one leading to World 6 shows her whacking said tower with a Parasol of Pain (complete with 1-up sound). Mario makes an approving noise and moves on, humming the theme song.
When you beat the first castle in World 8, it looks like you've rescued Peach... And then it turns out to be a cardboard cutout that flops over pathetically. The real humour comes in if you're watching without 3D on, as the resulting Depth Deception makes you think it really is Peach for a second.
It's even funnier if you spotted it using the binoculars earlier in the level. Even if you're playing with the 3D on, it really does fool you into thinking it's the real Peach.
Then Bowser flies away representing a certain internet meme cat...
Funny Background Event: As Mario and the Toads are flying home, Bowser tries to reach them with a Goomba Tower. They can't quite hold him...
The Propeller box and coin box powerups are pretty funny. You are just a box with legs running around!
ThisBrawl in the Family comic was made a little while before the game came out. One of the outfits designed, Tanooki Toad, is actually in the game In the ending credits and the Special World, although it does not look exactly the same. Also, one of the other Tanooki forms appears in the final letter...
Before it became canon, there was already fanart of Peach in a Tanooki dress.