Electronic – My DIY RGB LED panel powered by 9V battery works but certain colors like blue, white, and purple don’t work

It is highly likely that your red LEDs are of lower luminous efficiency than your white LEDs.

Also, driving an LED incorrectly can destroy it or reduce it's output from then on. Such damage can occur almost instantaneously. Red LEDs have lower operating voltages than White LEDs so are more likely to be damaged by the application of a constant voltage that is too high and not current limited.

The efficiency of LEDs varies widely. It can be expressed in lumens/Watt = l/W.

The best white and the best red LEDs have similar enough l/W ratings that both should appear about equally bright when driven with the same power input. White LEDs typically have operating voltages = Vf in the 3.0 - 3.6V range and red LEDs have Vf = 2.0 to 2.5 Volts. So a red LED would need about 50% more current to achieve the same power as a white LED as Power = V x I.

You state the LED output range as 500 - 20,000 mCd. That's a 40:1 brightness range.

Note that candela are a measure both of energy out and area of illumination. Less area = more brightness for the same light energy.

I think you mean that you will put LEDs in series (ie daisy chain strings) not parallel. if you use 3 times 9 volt batteries in series that gives you 21 volts . a serial string of 5 green, purple or pink should work well with a 200 ohm in series . for the other colours (2v lEDs) a string of 10 LEDs with a 100 ohm resistor can be connected to the 21 volts . you can make a 100 ohm by wiring two 200 ohms in parallel.

If you must have 5 LEDS in a string so that you can switch in groups of 5 you can run 5 (2v lEDs) in series with a 600 ohm resistor . 3 x 200 ohms in series = 600. The group of 10 is better because it wastes less power.

The above should work . If you need to accurately set the brightness . you would need to use a larger value resistor to make a string dimmer or a smaller resistor to make a string brighter.