How do devices/appliances draw more current when needed?
I decide to do something that requires more processing power. In this case my computer will draw more current to increase power.


Other way round: the computer will do more things, and as a result it will consume more power.


  Does my computer open up more parallel circuits so the total resistance decreases?


This is roughly true. Except that computers don't really operate on continuous current flow, they operate in bursts driven by their internal clock; each action involves either drawing some current to turn on a transistor, or sinking some current to turn it off again. Times a billion transistors, a billion times a second. More computation involves more transistors.