Preheat the oven to 500 degrees F. Set a wire rack in a rimmed baking sheet.
Cut slits into the branzino skin 1 inch apart, being careful not to cut through the spine. Stuff each slit with the garlic then drizzle the fish with oil. Season heavily with salt and pepper. Stuff the fish cavity with the lemon slices and the thyme sprigs. Transfer the fish to the wire rack and roast in the oven until the flesh is flaky and cooked through, about 14 minutes.
Remove and discard the thyme sprigs. Sprinkle with the parsley and serve.
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