The ability to bend space, “warp” does not actually have the ship travel any faster than it was already going. It might not be any more feasible than FTL of course because of the energy requirements, should we even conceive of how to do it.
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In my stupid little theory of time travel, we would actually be going to the “past” in a different time line, or another part of the multi-verse. Because who ever said parallel universes are all on the same date?
IOW, you could go back and kill your all your ancestors and you wouldn’t go anywhere but the you in that universe’s future would never exist.
I’m of the opinion that “parallel Universes” are not likely, but that’s my personal belief.