Thermostat/Furnace Wiring Confirmation
From your picture it looks like in reality that the yellow (Y) wire from the fatter thermostat cable connects to the red (R) of the smaller two conductor cable. All I can see is that the white (W) of the smaller cable connects to the COM (C) of the furnace board. (That wire may look yellowed because of age but I suggest that it is actually white).

The yellow (Y) wire from the thermostat is typically the conductor that is energized to call for cooling. From your picture it is almost assured that the two conductor cable goes to your air conditioning compressor unit to cause it to turn on. The return for the current going to the compressor has to return to the 24VAC and that is what is the white wire of the two conductor cable. 

The way that compressor control wiring is hooked up looks quite normal. 

You should be able to easily use one of the spare wires in the fatter cable to feed a common connection from the furnace back to the thermometer. The blue or black wire is often used for this depending upon what is in the cable bundle.