No doubt you’ve been told the simplistic “rule” that between describes the interaction between two parties and that once three or more parties are involved, among must be used. This may serve in the fifth grade, but we can deal with more sophisticated usage.
Here’s a fine example of the circumstance in which between can be used for more than two parties, from Mary Beard’s excellent SPQR: A History of Ancient Rome.
Describing the dynamic of the association of Julius Caesar, Pompey the Great, and Marcus Licinius Crassus in the First Triumvirate, she writes: “There were all sorts of strains, disagreements and rivalries between the three men. …”
That is, there may have been conflicts between Caesar and Pompey, between Pompey and Crassus, or between Caesar and Crassus, but all three men were severally involved in the dynamic, though only two at a given time may have been at odds, which makes between the apt preposition.