What happens when one resistor is used to limit the current through several LEDs connected in parallel?
The forward voltage of an LED varies between devices with temperature and with forward current. When connected in parallel, the voltage across each device is identical which causes uneven current sharing between the LEDs resulting from temperature and device variations. This results in brightness variations between the parallel-connected LEDs at best and, in extreme cases, can lead to premature device failure. Therefore, each LED connected in parallel should have it own current limiting resistor.