Why does LED dim when adding second LED in parallel with a single resistor
The voltage across an LED is about 2V and stays at just about 2V.

With a fixed power supply voltage it means that you have a fixed voltage across (and therefore a fixed current through) the series dropper resistor.

The resistor is acting as a constant current source and its current is shared between however many LEDs you have in parallel.

Lets say you have a 12V supply and a 1k Ohm series resistor. You then have 10mA through the resistor. With one LED there is also 10mA through this LED. Now if you add another LED there is 5mA through each LED and the brightness will drop noticibly. The current through each of the two LEDs is 50% what it was through the single LED.

Now adding a third LED the current through each LED drops to 3.333mA but the % drop in current is less than the drop when you added the second LED. The more LEDs you add the less the % drop in current.

So when adding more LEDs than two there will still be a brightness drop but it becomes less and less perceptible the more LEDS that you add.

Because the voltage which an LED 'wants' to have across itself varies slightly from LED to LED it means that you shouldn't run LEDs in parallel off a single series resistor. If you do it forces all the LEDs to have the same voltage across them which can cause differing currents between the LEDs. Some will have low currents and some will have high currents. Perhaps too high for there specification limit.