The link posted by Mike has a good explanation of the potential downside of using one resistor for multiple diodes. It will often work, but if one LED has a lower forward voltage than the others, it will conduct more of the current than the others, might be brighter than the others, and might fail sooner because it is conducting more current.
Not long ago, I conducted an experiment with 8 LEDs in parallel using one resistor. I set the voltage to a level where the LEDs were moderately bright and just let the LEDs glow. To my eyes, all 8 LEDs appeared to be equally bright. All was fine for about a hour, then one LED failed. The other 7 LEDs continued working for many hours before I stopped the experiment.
What did I learn? I learned that not all LEDs are created equal, even when they have identical specs and came from the same package. Apparently, the failed LED was out of spec compared to the other 7. It probably had a lower forward voltage than the others, causing it to conduct more current, resulting in a shorter lifetime.
It was exactly the sort of failure that can occur with multiple LEDs and one resistor. That does not mean that it will always occur. Your mileage may vary!
- Jeff