How do I control multiple 4 pin LEDs with minimum GPIO pins?

I'd like to mention Charlieplexing, which exploits the directionality of diodes and the ability for microcontrollers to both source and sink current to efficiently multiplex an array of LEDs.

I'm not sure exactly how well it will work with your common cathode RGB LEDs but it might be worth giving a try along with other methods

Use 2 Pcs HV5222PG-G, which is a 32-bit shift register ic. Or you can use any microcontroller to do so.

You will need 42+3 = 45 Pins in total.

You want to multiplex your LED's.

This http://lednique.com/display-technology/multiplexed-display/ (Simple introduction) and this https://www.instructables.com/Practical-Guide-to-LEDs-4-Matrix-Multiplexing/ (more in-depth) are what you want to do.

It's not complex or unknown. You just didn't know what to call it ;) You'll probably need some line drivers or (if you're cheap like me) a bunch of transistors because your CPU's I/O won't be able to directly handle the current. The switches shown in the referenced links are, obviously, transistors or drivers....