I suspect that Vonnegut is not so much fascinated with the question of parallel time as that he has chosen to use it as the perfect vehicle for this novel that is more a question of man’s warring nature and inability to change. Since the story encompasses the protagonist’s WW II experiences as well as his present state as an optometrist and some relevant backstory or flashbacks, the hopping around in time and space suits the story wonderfully well. Vonnegut also has hinted that Billy Pilgrim really isn’t a time traveler, but rather a dreamer and rememberer of experiences:
When Barbara left, slamming the door behind her, Billy traveled in time to thte zoo on Tralfamadore again. A mate had just been brought to him from Earth. She was Montana Wildhack, a motion picture star.
Montana was under heavy sedation. Tralfamadorians wearing gas masks brought her in, put her on Billy’s yellow lounge chair, withdrew through the airlock. The vast crowd outside was delighted. Everybody on the planet wanted to see the Earthlings mate. (p. 125)
The cash register where Billy waited for his chnage was near a bin of old girly magazines. Billy looked at one out of the corner of his eye, and he saw this question on its cover: What really became of Montanta Wildhack?
So Billy read it. He knew where Montana Wildhack really was, of course. She was back on Tralfamadore, taking care of the baby… (p. 195)
The name of the book was The Big Board. He got a few paragraphs into it and then he realized that he had read it before–years ago, in the veterans’ hospital. It was about an Earthling man and woman who were kidnapped by extra-terrestrials. They were put on display in a zoo on a planet called Zircon-212. (p. 192)
I believe that Vonnegut has exquisitely made use of the principle of time and space travel, its constant nature, its spectrum and uniqueness yet its common thread of human fallability to direct attention at the uselessness of war. His repetitious "So it goes" is one more sad commentary on the inability to change. The alien Trafalmadore society does not have war; it is just that–alien.