Help with powering LEDs in parallel with maximum brightness
The formula for a resistor for an led is:

(Source Voltage - Led Target Forward Voltage) / Target Forward Current In Amps = Resistance

Assuming 5V since you are using an Arduino, 20 mA forward current so ~2.2V forward voltage, that is (5 - 2.2) / 0.02A = 140 Ohms. So much bigger than your 470 ohm resistors.

You have two options. You can put multiple resistors in parallel (3 470Ω resistors in parallel is 153 ohms so roughly 18 mA, not even discernible to the human eye in brightness difference), or you can increase the source voltage ((12V - 2.2) / 0.02 = 490 Ohms, close enough). Or you know, just get different resistors.

As for your question on fading the leds (as a whole), there is plenty of examples online for Arduino led PWM. Connect all your parallel leds+resistors to a transistor or mosfet and use the Arduino gpio to PWM the transistor base/mosfet gate. Or get even fancier. Google Arduino led marque or led chaser for examples on how to make the leds chase each other around the heart.