Actually, the time travel and reincarnation theories could go hand-in-hand, it's the parallel one that is entirely different story. Although the latter seems more interesting, with two worlds co-existing at the same time, I really prefer our main protagonist (Jun) to remain as an original and/or special existence.
I also lean a little more with the former theory, because if we follow the pattern of the story, same people can't co-exist in the same world, that's why there's only 1 Jun, 1 Shiwoo/Myung, 1 prime minister and 1 Jin in the current world (for now, at least).
And if Jun was already dead, there's no need for him (or any other character) to exist in a parallel universe, because he is already dead in own world. The parallel world is not his world in the first place, and because his existence will bring anomaly to the "other" Jun in that world. Unless he time-travelled to the future, and met the reincarnations of the people he knew from the past.
Another thing, the concept of "Parallel Universe" is basically another world where another "us" exist, don't you think it's entirely pointless and contradictory if we die in our own world, only to be reincarnated to the parallel one? The whole thing denies the law of the concept itself.
Jin must be Jun's bro, who conspired with the Prime Minister dethroned Jun.
In Ch.7 Jun said "I definitely died from a sword." -- If Jun already died there and they already got the throne then they have no reason to follow him here..
That makes possibility of Parallel World or Reincarnation is higher than Time Travel.