They're allowed to because they bring in the ratings (and yes, these events are televised, otherwise the Lakitu Bros. camera guy wouldn't be there). Imagine how the Koopa, Boo, and Piranha Plant demographics would feel if they didn't have any representatives to root for. :P
but I still question the Party games(Did they really happen? Do they have alternate continuity? Is there a special Party dimmension, etc?).
That depends. All of them have plots in some way or another, but Mario Party 2's plot is revealed at the end to be a stage show at the Mario Land theme park. Mario Party 3's plot is debatable considering how it ends with whichever character you picked becoming "the Superstar of the Universe", but it's implied that Luigi is the default character (the cursor is on him at the selection screen, he doesnt have a corresponding Star Stamp, Waluigi is the villain, Waluigi's Island looks like it previously belonged to Luigi).
Mario Party 6's introduction is told entirely via the Miracle Book and is said to take place in "Mario Party World", so perhaps the events of that game are really just a written story. The rest of the Party games all fit pretty well with the exception of some odd out-of-character moments (really, Mario not being able to stand up to Bowser? Bowser suddenly turning nice for no reason? Mario and Wario not hating each other's guts?), though those could probably be explained away one way or the other (Wario could easily be an uninvited party crasher for instance).