I'm looking to recreate the line placed above the word "ZARA" in the following example.
Aside from using the Pen tool to get the shape, how can I change the stroke at both ends of the line?
You can use the Width Tool to vary the stroke of a path in Illustrator.
However...that is now how I would create that shape. It would be easier to just draw to ellipses and use Pathfinder's Minus Front to cut out the shape. To demonstrate, here's what you need to draw:
With both selected, using Minus Front from the Pathfinder pane will produce the final shape.