Gotta start off by saying that I actually really liked the concept of this animation. I liked Fairly Odd Parents as a kid, and like Team Fortress 2 now, so its a good sort of mixture of childhood nostalgia and jokes that I find funny now. You captured the art style of the cartoon and applied to the TF2 theme very well. There where nice little touches thrown in (like the teleporters, the pipe bombs) that made parody that much better. And I actually liked the "Hats" bit near the end. This was an idea that had a lot of potential, and for the most part, you used it. (Unlike a lot of parodies that get throw together these days) And that I commend you for.
Now to the negatives. The lyrics for the song just kinda fall apart in the middle, and never really find their way back to being that good. The first half is great, with lots of TF2 references that actually rhyme with each other. Then the "chorus" kicks in and things just stop working. The the "Red team and not the blue team" bit just feels really forced in there and breaks the flow of the song. It just sounds so weird. I know parodies are at there best when they reference the source (hehe) material but don't relay on it to make the supply the jokes, but really, it just seems like an unnecessary change that damages the final product as a whole. What was wrong with just saying putting something like "Odd Fortress, Fairly Odd Fortress" or even just "Team Fortress" there like the original theme song? You even did that thing where you stretched a word out or shrunk a word down so you could have enough syllables (A personal pet peeve of mine)
While I'm on the subject of out of place lyrics, I gotta say, the sound bites from the game don't do it for me either. The medic saying "I'm fully charged" sounds once, again, forced and out of place when put up against the original song. Even the spy at the end just seems like just throwing in a random sound clip. (Why it was the spy anyway, is another question, since at the beginning you established the blue solider as the Vicki character. [Sorry, I'm a continuity Nazi]) It just seems like you only put those in because there is some kind of internet law that you need a TF2 sound bite in a TF2 parody. I can understand not wanting to using a impersonator, and trying to keep the voices as close to the game as possible, I get that. But there are like hour long videos on YouTube of all the dialog of each class, and I'm sure you could find some quotes that would fit better than the one you chose. And even if you couldn't, you could take the Zeurel route and make your own sound bite with audio you have. Yes, it would take a lot longer and be a more difficult. But it would just be make the song that much better.
Which brings me to my last complaint. The voice. Good god the voice. I'm sure the actual recording itself was fine, but the filter you put over it just made it difficult to understand and horrible to listen to. If you didn't put the lyrics in the description, I would not have been able to understand a lick of it. I assume you put it in to try and make it sound like the original, multi-layered sound of the original, but in the end it just sounded like a really bad auto-tune job. Really took out a lot of enjoyment out of the cartoon.
That was a lot longer than I thought it was going to be. Whoops. Anyway, yeah this was a good idea for a parody, but there where just a few things holding it back from how great it could have been. When its good , its great. When its bad, its horrible. That being said, it is one of the better parodies that I've seen in a while, for the most part at least. If you do anymore parody cartoons in the future, just keep in mind somethings I mentioned and I'm they'll be even better.