It's logic. If I actually exist when you go back in time to kill my parents, it means you have already failed.

Let me explain it like this.
If I alter time by killing your mother, can I go back to when I hadn't had the thought of killing her, and her still being alive?

No.

Time alterations would be permanent. You cannot undo time.
Therefore, everything in a timeline is set in stone. Aterations simply make a new timeline.