Since you can't run more than one in series off that 6v supply (unboosted), then ideally go with a resistor each, as you correctly conclude if you used one resistor for all the LEDs in parallel, every LED that goes out, increases the current to the remaining ones, causing a cascade failure, eventually - not likely instantly.
If you can use say 12v, then six series strings of three with 1 resistor per string would be a good compromise.