How to make an edge loop that is parallel to one of the sides of the face loop?
Not this -
But this -
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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 > 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.