Why must resistors be on the respective anode terminals instead of the common cathode terminal of a RGB LED?
The red led will hog all the current because it might only need 2 volts across it to begin conducting. The green and blue LEDs need a higher voltage but because they are all in parallel the red led dominates. Try measuring their respective volt drops when each operates.

It's like putting a 5 volt zener in parallel with a 10 volt zener. The 10 volt zener will never conduct.