In parallelism, mental events and physical events are perfectly coordinated by God; so that when a mental event such as Sally's decision to walk across the room occurs, simultaneously Sally's body heads across the room, in the absence of a direct cause-effect relation between mind and body. Mental and physical events are just perfectly coordinated by God, either in advance (as per Gottfried Leibniz's idea of pre-established harmony) or at the time (as in the Occasionalism of Nicolas Malebranche).
Parallelism is a theory related to dualism which suggests that although there is a correlation between mental and physical events there is no causal connection. The body and mind do not interact with each other but simply run alongside one another, in parallel, and there happens to be a correspondence between the two but neither cause each other. That is to say that the physical event of burning my finger and the mental event of feeling pain just happen to occur simultaneously - one does not cause the other. An example given by Janice Thomas is one of a punctual student who always arrives on time for his classes. One of his classes begins at four, so whenever he arrives in the room for that particular class a clock strikes four. However, it is not his arrival that causes the clock to chime - the two just happen to coincide. An appeal is made to God, as it is believed that at the beginning of time, God sets up a mental chain of events and a physical chain of events and ensures in his construction of them that they will always run in parallel.
In his book The Mind and its place in nature, C. D. Broad maintains that parallelism is, "The assertion is that to every particular change in the mind there corresponds a certain change in the brain which this mind animates, and that to every change in the brain there corresponds a certain change in the mind which animates this brain."
- Broad, Charlie (1925 reproduced online at http://www.ditext.com/broad/mpn/mpn3.htm). The Mind and its place in nature.