Why wasn't Johnny Lawrence (William Zabka) in Karate Kid Part III? Zabka played the character in the first two Karate Kid movies, and is currently starring in the YouTube Premium series, Cobra Kai, but he was absent from the third installment in the original trilogy.
In the classic coming-of-age martial arts film from 1984, The Karate Kid, Daniel LaRusso (Ralph Macchio) is bullied by Johnny Lawrence, forcing him to learn Karate from Mr. Miyagi (Pat Morita). The movie culminates in an iconic final fight between Daniel and Johnny at a martial arts tournament. The fight ends with Daniel claiming victory and Johnny bringing disgrace to the Cobra Kai dojo. YouTube's Cobra Kai series brings back both characters, and focuses on a middle-aged Johnny's efforts to reopen Cobra Kai.
Johnny has been in most installments in the Karate Kid franchise (including Cobra Kai), but wasn't in the third movie, which saw Kreese (Martin Kove) try to take revenge on Daniel by training a new opponent for him to fight. Johnny wasn't involved because his story had already been told in the first film. Though it's true that he appeared in the second movie, his role was brief. The Karate Kid Part II opens with the immediate aftermath of the tournament and shows Kreese's angry, violent reaction to Johnny's loss. In this scene, Mr. Miyagi acts as if he's going to attack Kreese in retaliation but tweaks his nose instead. This scene mirrors how Daniel will deal with Chozen (Yuji Okumoto) at the end of the movie. Another possible reason for the scene is that it sets up the plot of The Karate Kid Part III by focusing on how the loss affected Kreese.
So in a way, Johnny Lawrence's role in The Karate Kid Part II was more about Kreese, Daniel, and Mr. Miyagi than it was about Johnny. The Karate Kid trilogy was essentially done with Johnny after he was defeated by Daniel in the first movie, and there was no need for him to come back into the story. Daniel had already proved that he could beat him, so the next Karate Kid movies had to move on to new opponents who could challenge Daniel.
So why did Johnny Lawrence return? Cobra Kai was envisioned as a show that looks at the characters of The Karate Kid from a different perspective, specifically the villain's. Zabka has said that when Cobra Kai was pitched to him, he liked the approach that they wanted to take with Daniel and Johnny (via Rotten Tomatoes). It was important to him that Cobra Kai was a "true continuation of that story". So even though The Karate Kid properly ended Johnny's story, Cobra Kai found a way to revisit the character in a way that was deeply revealing and nuanced.