These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Accidental Innuendo: In the anime, Migi saying he'll find a way to make Shinichi feel better... while they're in the shower.
Base Breaker: Murano has the anime's fanbase split; being seen as The Scrappy for some fans. While she does still have people that find her character justified considering the situation she's in regarding her feelings for Shinichi, detractors dislike her for her negativity towards Shinichi's change from kindhearted Nice Guy to Jerk with a Heart of GoldBadass, and her tendency to repeatedly ask, "Are you really Shinichi?" making her come off incredibly annoying. It doesn't help that she's more demure and less feisty in the anime compared to the manga. This may be a downplayed example, though. While these fans tend to hope Kana wins Shinichi's heart, if only so he never goes back to being a Nice Guy, very rarely do any of them actually wish for Murano's death.
The anime's soundtrack in general, but especially its use of dubstep.
The artstyle and character design changes and setting update have also been divisive among fans of the manga. Some accept them, some don't.
Complete Monster: Uragami is a human being who manages to be many times worse than the man-eating parasites of the story. A sadistic Serial Killer, he considers other people his "playthings" and gets off on all acts of cruelty he can inflict them, including murder, rape, mutilation and cannibalism. After he was caught and forced to work for the army, he gleefully helped them slaughter dozens of parasites along with a few innocents before killing the police officer escorting him and escaping. Knowing that it was only a matter of time before being caught again, he decides to spend his last days as a free man forcing protagonist Shinichi, who had never done anything to him, to confront him. He does this by killing more innocents, capturing Shinichi's girlfriend Murano and threatening to "have a lot of fun breaking this little doll" if he doesn't comply, only to throw her from a rooftop and stabs Shinichi when he tries to save her. A depraved fiend, Uragami was even able to see through the parasites' disguise because he was, like them, a monster hiding behind a human appearance.
Fandom Rivalry: Parasyte vs Tokyo Ghoul can be quite a heated debate at times. Both have relatively the same dark-toned setting, and their anime adaptations were released around the same time as each other.
Fan Nickname: After the life-saving surgery below is explained in episode 7, the concept of Migi's cells affecting Shinichi's body became known as "Migi-chlorians". The resulting drastic Art Shift for Shinichi also gave rise to the term "Shinichi 2.0".
Thanks to the hero's surname being Izumi and Aya Hirano's involvement, there's a lot of fanart with Lucky Star.
Narm: When Shinichi is at death's door, Migi fills the roles of trauma surgeon, nurse, AND organ donor in order to save his life. It should have been the dramatic turning point of the story in more ways than one, but thanks to the visuals that accompany this scene... not so much.
Even more so in the anime, where it is voiced by Aya Hirano.
Averted, however, in the live-action movies, which give Migi a very deep voice and its creepiness is turned up a notch.
Uncanny Valley: In the story, Parasites generally are indistinguishable from normal humans. However, they tend to have completely blank expressions due to not having emotions, and the artwork often gives them slightly overlarge and sharply-defined eyes that give them a sinister appearance. Extremely twisted humans, such as Uragami and Takeshi Hirokawa, also have these eyes. There's also Miki.
What an Idiot: Episode 9 gets this with Shinichi's classmate, Yuko Tachikawa. Instead of informing anyone, or even her brother who just so happens to be a policeman, about discovering that Hideo Shimada (a Parasite) isn't all that he appears to be, she decides to confront him on her own. Yeah... great choice, Yuko. Doesn't help that this action ended up getting 17 people killed by said Parasite.