Using a handheld slicer such as a mandoline, thinly slice zucchini lengthwise. Choose the 24 widest slices and save the rest for salad. Using a vegetable peeler, make 24 parmesan shavings the size of the zucchini slices. Discard spinach stems, stack leaves, and cut into thin ribbons.
Lay 4 zucchini slices flat on a work surface. Smear each with 1/2 tsp. tapenade. Put a parmesan shaving on top; it’s okay if it breaks. Set a generous pinch of spinach at 1 end of each slice. Roll up from spinach end, close with toothpicks if you like, and set on a platter. Wipe off surface and make rest of rolls the same way.
To serve, drizzle rolls with olive oil and sprinkle with pepper.
MAKE AHEAD: Through step 2, chilled airtight up to 3 hours. Lift rolls and blot moisture on platter with paper towels to continue.