Keep in mind, this game is nothing but stupidity and jokes. A giant joke.
That is not a parody.
While parody often uses humor, humor is not parody.
Parody is imitation of a thing, with exaggerated effect, that makes a statement about the thing itself. The law requires you to keep the use down to a minimum, using only enough to convey the message.
Several web sites to a great job of this.
As an example, one video clip used Mario and other characters. Instead of the mushroom kingdom being invaded (as the original SMB games portray events) they are reinterpreted: The scene of SMB level 1.1 is the same, with Mario initially stomping on one goomba, then aggressively stomping on two goombas, but we discover it is not the mushroom kingdom, it is actually Bowser's kingdom! Soon the game progresses with actual scenes, Mario is destroying the kingdom in a rage. Meanwhile Bowser has not kidnapped Princess Peach, but the two of them are together in his castle working out peace accords between their kingdoms. The are using the characters, but doing it to make a statement about the product itself.
The video clips for How It Should Have Ended (HISHE) usually do a good job of this, but have been bitten legally.
Generally in HISHE they use just enough of the characters and script to show you a scene, but then throw in a twist that would dramatically end the movie. That is very much a parody because they are making statements about the movie, or sometimes making statements about the actors or directors or writers. Often the clips show what would have been intelligent decisions that would have avoided most of the conflict. However, they were bit hard in a few of their clips when they went beyond parody. One of the biggest was their "Frozen" clip since they wanted to include a single verse of the "Let it Go" song. The song was NOT a parody since it was not necessary to make a point about the movie. They were contacted by legal teams, told it constituted a performance of the song and was not fair use. They took down that version of the clip, wrote a new song that did make a statement about the characters, and re-issued the revised version.
The "Honest Trailers" group similarly uses parody, taking clips from the movies but overlaid with a different speaker in order to make various statements about the movie itself. That is a standard use of parody.
Not all parody is comedy. There have been many parodies that are quite serious, parodies that are dramatic, and parodies of serious situations that make statements showing how terrible the problems really are but how most of society misses the details.
If you are just using the things because they are funny, that is NOT parody. It routinely gets shut down. Since you say you want to use "numerous companies and characters" you are basically opening yourself up wide for every one of them to take legal action. And based on your description, they will quite likely win against you.
And as written above, fair use is an affirmative defense. If it actually goes before a judge, you are effectively telling a judge "I know this would normally violate the law, but I believe it is exempt. I have made sure to only use the minimum amount possible, the entire use makes a statement about the item itself, and I don't use the materials for other reasons." If the judge disagrees with the fair use assessment, you have fully admitted to using the product and will need to make whatever reparations the court sees fit.
