18650 battery pack. Current distribution
I'm thinking that the 80A current going to the battery cable will have
  to pass through a single point on the battery which can currently only
  handle 13A (1 parallel branch)


Depending on where you connect the output leads, that could be correct. Even if you connect it to one of the inner cells the current going through straps in adjoining cells will exceed their rating, since it will be the sum of cell currents in that leg.  


  So I'm not sure if we should have 6 wires connected to each parallel
  cell to distribute the current before connecting to the main battery
  cable.


Yes, that is the best solution. Each wire only has to handle ~13 A so it can be a thinner gauge, which is easier to handle than a single wire rated for 80 A. If the wires are all made the same length then they should distribute the current more evenly between cells.