Tha amp is drawing excessive current and using all the available voltage it can. Your alternator is outputting approx 14.4 volts. When the source unit is turned up and the amp is playing at loud volume, it needs more power. So now that the amp is needing more power, it takes what it can, and the voltage available to other components diminishes. When this happens, other accessories don't work to full capacity, such as your lights. A lot of times, a capacitor will help, because it will stablilize the voltage. So instead of having the voltage fluctuate from 14 volts to 12volts for example, the electrical property of a cap will stabilize it to whatever the nominal incoming voltage is.
You can also help this problem to some degree by replacing the factory wiring, especially the ground cable from the battery to the chassis and making sure you have 0 ohms from the chassis to your amp's ground terminal.