Honeywell Zone System Wiring
Remove the jumpers, they're superfluous

It appears from the installation manual for your zone controller that the C terminals for the thermostats are connected internally to the C wire coming into the board from its 24VAC power source.  This means that the white jumper wires between the zone C terminals and the blue C wire from the furnace can be removed altogether and your C wires hooked up in their place.

As to the 1K resistor?  Someone thought this unit power-stole, I guess, and put that in there to provide a bypass load to prevent erratic operation.  I'd remove it and keep an eye on the system -- if it doesn't malfunction after a week or two without it, I'd say whoever put it in was a member of a power-stealing cargo cult.