Different coloured leds have different forward voltages. Red leds have the lowest, usually about 1.8ish volts, while green and blue are closer to 2.5v. So if you have for example a red led in parallel with a green one, the voltage across both the leds will be the forward voltage of the red led, 1.8v. Too low to turn the green on.
If both the leds had the same forward voltage, both of them would light up. If you want both of the different colours to light up, you should use a separate resistor for both of them