Why did Arren kill his father? This was the biggest question I had for Gorou Miyazaki's debut animated film Tales from Earthsea. Was the motivation for Arren's murder of his father ever revealed in the anime or in the original Earthsea series by Ursula K. Le Guin? Or was it just a projection of the inner feelings the director had toward 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