The first act is a parody of the old cartoons like Mickey Mouse from before color TV was invented.
At one point, to get stronger muscles, Fry eats jelly from a can, and his muscles swells up like mushrooms. This can be a reference to how Popeye the Sailor used spinach to make himself stronger
At the end of this segment, Bender bursts through the screen saying, "T-t-t-that's all you get, ya jerks!". This is a reference to the end of most Looney Tunes shorts where Porky Pig bursts out of a drum saying, "T-t-t-that's all, folks!"
Second segment:
The second act is a parody of many video games.
In the second segment, when the professor determines the meaning of everything, it seems to parody art games like Every Day the Same Dream and One Chance.
After the professor has solved all the questions of the universe, there are many references to various video games: Space Invaders, Boulder Dash crossed with Digger, Donkey Kong, Tapper, and Dig Dug, plush the clouds and pipe from Super Mario Bros. 2 (Japan only version).
Third segment:
The third act can be a reference to Japanese animation shows.
Amy's hairstyle in the third segment seems to be a parody of Sailor Moon.
In the third segment, some of the music is actually from real anime shows, esp. Voltron and Macross.
Bender's appearance resembles an RX-78-2 Gundam in the popular space opera mecha anime series Mobile Suit Gundam.
Bender transforms into a bigger and more powerful robot, which can be a reference to the many continuities in the Transformers franchise.