Parallel circuit not working as expected
This idea:


  From what I have learnt in physics I thought that when a circuit was in parallel the components acted as if they were independent of each other however this is not what I see here.


is based on the assumption that the power supply you are using is more than capable of driving all the items in parallel. The battery has an internal resistance, and as the current through it increases, the voltage out of the battery drops.

You are likely pulling too much current out of the battery for the LEDs, and as such the voltage from the battery drops to a level that is too low to power the Arduino.