Why did Arren kill his father?
Gorou Miyazaki had denied that Arren's murder of his father was a reflection of his own feelings; rather, it was more or less the feeling of the young Japanese generation:


  On Earthsea, Goro denied that Arren’s oblique patricide reflected the director’s own feelings in an interview at the Venice Film Festival. “I do not have much relationship with my father; because of that, I have never felt like killing him. I decided to start with the son murdering the father because I understand that’s more or less the feeling of the young Japanese generation. When I worked in the Ghibli museum, most of the staff there were young people with common problems. I wondered about why that was and tried to come up with reasons, which are reflected in the film. I purposely didn’t explain why Arren stabbed his father because I wanted the audience to think about it, and reach a broader idea of why these problems exist.”


Source: "Tales from Earthsea and family feuds", UK Manga Entertainment Website

However, as Gorou purposely didn't explain the why, it proved to be a challenge for many viewers to figure that out for themselves. For what it's worth, Ursula K. Le Guin thought the murder was committed without much reason:


  The moral sense of the books becomes confused in the film. For example: Arren's murder of his father in the film is unmotivated, arbitrary: the explanation of it as committed by a dark shadow or alter-ego comes late, and is not convincing. Why is the boy split in two? We have no clue. The idea is taken from A Wizard of Earthsea, but in that book we know how Ged came to have a shadow following him, and we know why, and in the end, we know who that shadow is. The darkness within us can't be done away with by swinging a magic sword.


Source: "Ursula K. Le Guin: Gedo Senki, a First Response", ursulakleguin.com