Preheat oven to 400 degrees. Cut out four 12-by-17-inch pieces of parchment; fold each in half crosswise to form a crease, then open.
Pulse parsley, garlic, and oil in a blender or food processor until a paste forms. Season each salmon fillet with 1/8 teaspoon salt and the red-pepper flakes.
Rub fish with paste. Divide spinach and chickpeas evenly among 4 parchment pieces, arranging mixture on 1 side of each crease to form a rectangle the size of a salmon fillet; season each pile with 1/8 teaspoon salt and some pepper. Lay 1 fillet on top of each. Fold parchment over ingredients; make overlapping pleats to seal.
Bake on 2 baking sheets (2 packets per sheet), 9 to 10 minutes for medium-rare or 11 minutes for medium. Unwrap.