Managing circuits beyond a basic level
The current through each LED would not change.  Each one would draw 5/330 or 15mA (I=V/R).  The power supply would have to supply 4*15mA or 60mA.  You didn't change the amount of current through any of the LEDs, all you did is add more of them in parallel.

Let's use a classic water analogy: you have a bucket of water with a single hole drilled in the bottom.  The amount of water flowing out of that hole  will not change if you drilled three more holes in the bottom.  The amount of water flowing out of the bucket, however, would quadruple.

Remember that in any circuit the current flowing through components in series will be the same, and the voltage across components in parallel will be the same. I think this is where your confusion lies.  Adding an LED across a power supply doesn't affect the voltage or current for the already-existing circuit just the same as plugging a radio into the outlet that a lamp is already plugged in to does not make the lamp get dimmer.

If you are still confused please comment and I will clarify. There's nothing wrong with wanting to learn. :-)