“Paradoxes are yet another reason for why time travel seems like science fiction and is easily dismissed as a real possibility by most people. However, there is a work around to this and it involves parallel universes and alternate timelines. Specifically, some theoreticians speculate that if you could indeed manage to time travel you would not be part of the same universe you left from, but rather create an entirely different one upon your arrival.”
(via Physics 101: Understanding time travel and why we shouldn’t dismiss the possibility)
These are the sorts of things that made my head hurt while writing my time travel romances, Highland Passage and Knight Errant.