I copy and paste @xmag's comment here. Both theories make sense
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Great drama, I wonder if it takes place in the same world as "First love again"? Because in this drama, an old cell phone was also the mean to travel, either time-travel or travel to an alternate universe. And in one episode, the first traveler (a monk from the 17th century) told him that many people were time-traveling, by different means (watches, phones, clocks...), so I wondered about that.
Also, I don't know, I doubt the traveling to an alternate universe (I mean, a parallel universe). Because it looks like the ML is alive in the end, with the umbrella scene. Also, in episode 19, the FL realizes that all her "travels" took her to a moment in time in the life of the ML, about what he wanted to do with her. It can't be a coincidence.
So I think that the travels took the FL into the ML's brain, while he was in a coma after his fall. This alternate universe is not a parallel one, one which would exist next to ours, but an alternate one he has created while in a coma, to relive moments with the FL.
Somehow, the FL was able to get into his mind, maybe after her wedding was cancelled and especially after she gave up for good on her cheating fiance. Her mind was free from her obsession with him and with the phone making the link, she could travel to the lives the ML was creating for them.
Maybe I'm wrong but I think it will be my headcanon, rather than the alternate universe or a parallel universe a la Fringe (the american tv series, great one, by the way, I recommand it).