Hello Ben --
I am not intimately familiar with Plymouth locomotives, but I think I can safely say this is some sort of homemade arrangement. As built, Plymouth locomotives of this vintage had Climax or Buda gasoline engines with a manual transmission fitted with a transfer case in the drive train that allowed full-on reverse operation. It looks to me like whoever replaced the original engine also had to make some sort of modification to the way it was mated to the transmission, so they included this extra shaft and chain drive. If your locomotive now has an automotive-type automatic transmission, I'll bet that's the reason.
Cheers -- Wayne