Prepare a grill or grill pan for high heat. Place the sugar, vinegar and 2 tablespoons water in a small bowl and microwave until warm, about 1 minute (see Cook's Note). Add the shallots and chile to the vinegar and set aside for at least 10 minutes.
Toss the squid with 2 tablespoons olive oil, salt and pepper in a medium bowl. If using an outdoor grill, place the squid in a grill basket and press together to flatten. Alternatively, for indoor grill pans, weigh the squid bodies down with a cast-iron skillet or bacon press. Grill the squid until cooked through, about 2 minutes per side. Slice the bodies into rings and divide the rings and tentacles evenly among 4 shallow bowls.
Place the parsley in a medium bowl and drizzle with the remaining 2 tablespoons olive oil. Toss with the shallots, chile and vinegar. Season with salt and pepper. Mound the salad on the squid and spoon any vinegar remaining in the bowl over top. Garnish with lemon wedges and serve.
You can also place the sugar, vinegar and 2 tablespoons water in a small saucepan and stir over medium-high heat until the sugar dissolves, about 5 minutes. Remove the mixture from the heat before adding the shallots and chile.
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