Please keep in mind that LEDs are current devices, not voltage devices. It takes a slightly different mind set when thinking of LEDs. Sure, thinking about them using voltage is possible, but it can also lead to problems like you are experiencing and being able to explain why something is happening.
In short, what you are seeing is as the LEDs blink on one side, it is loading the system down more, causing a slight voltage drop in the wiring associated with the other side (since they use a common supply). This causes the LEDs to have a different current flowing through them (the intensity of an LED is related to the current supplied to it, which also affects the voltage drop across the LED). So, as the voltage drops, the LED pulls less current, therefore you have a slight change in the intensity of the lighting. Most of this voltage drop is more than likely coming from your flasher module. I would try a different brand and see if that helps any. It should. Otherwise, you are looking at having to upgrade to a larger gauge wire to solve this issue. Once you get beyond the flasher, then you do not have to worry about this.
Please keep in mind that if your tail lights are going to a common ground point, that can also be a source of your problem due to the voltage across the LEDs would be a function of the supply voltage (current) minus the drop across the high resistance connection. As you turn on more LEDs (with the flasher), you have more current flowing through the ground wiring, leading to more losses.
I am trying to keep this simple, but it takes a certain amount of electronic understanding to really explain this.
