Cut off and discard any dried section at the bottom of the asparagus stalks—super fresh stalks may not require trimming at all, but most benefit from being rid of the bottom inch or so. Chop the trimmed asparagus stalks into 1- to 2-inch pieces; if you want to have some more texture in the final soup, set the tips aside separately.
Rinse and drain the chickpeas. Put the chickpeas and the broth in a medium saucepan. Bring to a boil over high heat. Add enough salt to perk up the flavor of the liquid; this will vary greatly depending on how much salt is in the broth you're using.
Add the asparagus (not the tips if you want that extra texture). Adjust the heat to maintain a steady simmer and cook until the asparagus is very tender, 5 to 10 minutes depending on how thick the asparagus stalks are.
If you have a hand-held blender, now is a great time to use it—whirl the soup in the pot until smooth. If you don't have an immersion blender, transfer the soup to a blender or food processor, whirling in batches, if necessary. Be sure to hold a kitchen towel over the blender to prevent hot splatters. However you blend it, whirl the soup until utterly, completely smooth.
Return the soup to a simmer over medium-low heat. Add the reserved asparagus tips and cook until the tips are tender, about 3 minutes. Add salt to taste.
Serve the soup hot, with a drizzle of the garlic-parsley mixture swirled on top, if you like.