Neither of these terms, "split personality" and "parallel reality," are precisely correct. The concept Kotzwinkle is trying to put over is a complex philosophical and psychological one: What happened to humanity when it is strained from within? and, What happens to personal identity when it is strained from within? Let's look at what is not the case with Caspian so we can better see what is.
Multiple personality disorder, "split personality," is a psychological event that is the effect of a traumatic cause: some horrible, life shattering event(s) occurs that fixates psychological and cognitive functions at the moment(s) at which the horror(s) occurred, "frozen," as it were, in that moment. This results is a narrowing of the range of cognitive process and emotional reaction when cognition and emotion are governed by those dual fixations. Caspian suffered no such trauma; his experience was an individual inner one in which he strove to access the deepest parts of his psyche for the sake of his performance.
Parallel universe/reality is a concept in quantum cosmology wherein it is theorized that for every option not chosen at a decision-making moment, an independent, separate (parallel) reality materializes that expresses the life situation that would correspond with those options not chosen while "present" reality corresponds with the option that was chosen: a world exists for what you did not do as well as for what you did do. Caspian isn't experiencing a parallel reality built on and corresponding to an option for an different decision in present time that he did not make.
What Caspian is doing is accessing his deepest psychological motives and psycho-cognitive processes without understanding of or experience of the mechanisms at work therein. Thus he is said to have created a psychologically based alternate reality. It is an alternate rather than parallel because it is a different time period and location with different relationships with the people in his present-reality life, each individual is there but in a different role, for example, his daughter is, in Berlin, a young girl in trouble whom he seeks to rescue. Caspian's adventures in his alternate reality go full circle and he is said to have "died" at the hand of the Nazis in 1940s Berlin while Felix Falkenhayn emerges into Caspian's 1980s life.