Question: Is it true that Bill is responsible for the murder of Oren-Ishii's parents?
Answer: It has been speculated for years, including on this site, that the sword-wielding killer in the anime sequence was actually Bill, but with the recent release of Kill Bill: The Whole Bloody Affair, it's now clear that that particular character is someone entirely different. Nothing in the movie suggests that Bill had anything to do with O-Ren's parents dying.
Question: During the anime scene, is the sword wielding killer that finally finishes off O-Ren's father supposed to be Bill? Could this be the connection between Bill and O-Ren?
Answer: Bill was behind the killing of O-Ren's parents, we think. It is never clarified.
Answer: Now that "Kill Bill: The Whole Bloody Affair" has been released, this question can be answered. (Super minor spoiler ahead) I also thought that this character could possibly have been Bill, as they are shown to hold their swords similarly when not using them, plus the hair and the jewelry. But Tarantino has added an additional scene, showing that that character is not Bill. He is referred to as Matsumoto's second in command, Pretty Riki, and is killed.
Answer: Yes, it's confirmed it in a post-release interview.
Grumpy Scot