Basically, you're filled with a magic juice that gives you the power to bend the universe to your will in overly and unrealistically fantastic ways, thus making you far superior to regular mortal beings. Now you must venture to a magical school, where you will undergo rigorous training, which involves risking your very life in ways no sane person would ever put their child through. Think you can handle that?
Alternative Character Interpretation- (In universe) Harry is horrified by the existence of magic, Ron is a complete and total moron, and Dumbledore is some kind of malevolent puppetmaster, just to name a few.
Death Is Cheap - In this version of the Potterverse, getting killed is nothing more than a good sign that you'll be resurrected by and join Professor Squirrel's team.
Dead Pet Sketch- A bit of a subversion in that Harry kills Scabbars but tells Ron it was Crookshanks, rather than pretending Scabbars is still alive. Also it was heavily implied the rat had been dead for years. Harry just disposed of the corpse because it was rotting too much. The next episode has Harry finding a "replacement" Scabbars, which Future!Harry already confirmed to be Peter Pettigrew.
Ron: Hey, Harry, guess what this is? Harry: A porcupine in a sweater. Ron: No, silly, it's a letter for me!
Evil Plan- Voldemort has had a few, ranging from throwing Harry off of his broomstick while he's playing Quidditch (which was seriously from the original book. What the heck?!) to sneak attacking him from a distance. Harry lampshaded the lameness of the latter plan in episode 15.
Never Say "Die"- Parodied and inverted in that while characters can die, everybody seems to believe that they have been expelled. Everybody except Harry, that is.
No Indoor Voice- Ron, which gets in the way when they're trying to be sneaky in episode 12.
No Such Thing as Wizard Jesus- Averted in the Christmas Special, in which Ron states that Jesus was a normal wizard, to the shock of Harry, Hermione, and Aberforth Dumbledore. Harry claims that Ron offended everyone, not just Christians, everyone.
"Not Making This Up" Disclaimer- Invoked at least twice. Once when the Chamber of Secrets opens for random sounds (like in Deathly Hallows) and once when Ron thinks the sleeping Lupin might be dead (in the parallel scene from Prisoner of Azkaban).
Noodle Incident - Getting past Fluffy (twice!) and the "forcefield rule" are never explained or mentioned again.
Out-of-Character Moment- Happens when Ron takes offense at Malfoy calling Hermione a Mudblood in Chamber Pot episode 6, which would be in-character in the original book, but Ron acts completely different in the parody. This is immediately lampshaded.
Overly-Long Gag - In Prisoner of Azkatraz, every time someone says "Sirius Black," [[Phantom of the Opera]] music plays. Lampshaded by Harry in the last episode, where he gets fed up with it.
Doctor: Headmaster, there's been a Transfiguration accident. We have another 500 students injured. Dumbledore: Eh, the Hospital Wing's full; just dump 'em in the lake. Doctor: The lake is also full, sir. Dumbledore: *indignant* Is the giant squid not hungry?!
Ridiculously Human Robots - The Defense Against the Dark Arts teachers. Also implied for other Hogwarts staff, the Muggles/copy-paste Hogwarts students, and Harry.
Sadist Teacher- Snape, who actually dangles Harry over a boiling cauldron with a rope. Dumbledore comes into the room and, of course, notices nothing out of place.
School Uniforms are the New Black: Regardless of whether or not they're in class, students perpetually wear these, even to bed at night. The only exception is when they're outside of Hogwarts.
Also applies to Harry, Voldemort, and Ron's parents.
We All Live in America - Parodied with the Prime Minister. Every time he speaks, the US presidential anthem plays.
What Happened to the Mouse- The scene where Ginny steals Tom Riddle's diary back from Harry was cut, so it looks like it was just forgotten about. This is immediately lampshaded once it becomes important.
White Void Room- One serves as the entrance to Dumbledore's office. The only feature is a door.
You ALL Look Familiar- Essentially all background characters (especially students, even if they had names in the original books) look exactly the same. The best example is the Quidditch team, where even though all members of the Gryffindor team had names in the original book, nearly half of them are swapped for the generic model, named "Muggle" according to the credits.
You Mean X Mas- Inverted in that while the wizards do celebrate Christmas, they seem to believe for some reason that it celebrates the birth of Merlin and not Jesus.