Why do different colored LEDs interfere with each other when connected in parallel?
The red LED has a much lower voltage drop for a given current. In that way, the red ones light but the rest don't achieve their voltage necessary to light. 


Red LEDs have a voltage drop of about 1.8V.
Blue LEDs have a voltage drop of about 3V.  


You can see more colors and their corresponding voltage drop here in this table: http://en.wikipedia.org/wiki/Light-emitting_diode#Colors_and_materials

To solve this issue, you need a separate current limiting resistor for each led.

You could think about it as if you were putting two different zener diodes in parallel. If you have a 2 volt zener and a 5 volt zener, the 2 volt zener will reach it's voltage and prevent the 5 volt zener from ever passing any current.