Trim the ends of the zucchini and slice into 1/4-inch half-moons. Thinly slice the onion and finely chop the parsley.
In a large bowl, whisk the eggs with the Parmigiano-Reggiano, kosher salt, black pepper, and parsley until smooth and lightly foamy, 30–45 seconds.
Heat the olive oil in a 10-inch nonstick or well-seasoned oven-safe skillet over medium heat. Add the garlic clove and sizzle until fragrant, about 1 minute; remove and discard the garlic.
Add the onion and cook, stirring occasionally, until softened and translucent, 6–8 minutes. Add the zucchini and a small pinch of salt; cook until tender and most moisture has evaporated, 8–10 minutes. Spread the vegetables in an even layer in the skillet.
Pour the egg mixture over the vegetables. Stir gently with a spatula for 20–30 seconds to distribute, then reduce heat to medium-low. Cook without stirring until the edges are set and the bottom is lightly golden, 6–8 minutes; loosen the edges all around with a spatula. If finishing under a broiler, heat the broiler now.
Finish the frittata: Place a large plate over the skillet, invert to flip the frittata onto the plate, then slide it back into the skillet to cook the second side until just set in the center, 2–3 minutes. Alternatively, place the skillet under the broiler 4–6 inches from the heat and broil until the top is just set and slightly puffed, 1–3 minutes.
Slide the frittata onto a cutting board and let rest 5 minutes. Slice into wedges and serve warm or at room temperature.