Parallelism can be seen as one of many attempts throughout the history of philosophy to answer the so-called mind-body problem: how is it possible for there to be any kind of meaningful interaction between mental and physical states? The two deal with entirely different things, the mind with thoughts and the body with physical movement. And yet somehow there appears to be some kind of correspondence between mind and body. They may be different, but they are not entirely separate.
Parallelism attempts to answer the mind-body problem by arguing that there is a harmony between mind and body. This isn’t the same thing as saying that the mind and body causally interact. In other words, the mind cannot cause the movements of the body, nor can the body cause thoughts in the mind. Properly speaking, causation only relates to physical objects, so this rules out the mind as having any causal influence. Instead, so the argument runs, the mind and body operate on parallel lines, giving the false impression that there’s causal interaction between them.
That the mind and body should appear to interact in the way that they do is entirely down to the pre-established harmony of creation set in place by God. Everything in God’s creation is harmonious, including mind and body. The philosopher Leibniz held that God had arranged the universe to look as if there was some kind of interaction, as a way of proving order and stability to the creation.
Outside theism, then, parallelism has no real meaning. It was a way for theists in the Age of Reason, theists such as Leibniz, to reconcile their belief in God with the existence of a physical universe that was becoming increasingly amenable to causal explanation.