How to make an edge loop that is parallel to one of the sides of the face loop? How to make an edge loop that is parallel to one of the sides of the face loop?

Not this - 

But this -
If you need a new loop cut:
In edit mode, press ctrl R to add a new  loop cut. Then during edge slide press E which aligns the new loop to one of the adjacent face loops. F switches between face loops.
If a new edge loop is not needed:
Then select a edge loop alt  RMB. Press ctrl E &gt; Edge Slide to start the edge sliding, then during edge slide press E which aligns the new loop to one of the adjacent face loops. F switches between face loops.
These are the options for edge slide as shown in the 3D view header.