Place potatoes in Dutch oven; add water to cover and 3 tablespoons sea salt. Boil potatoes until tender; drain and cool. Cover and chill several hours or overnight.
Combine garlic and next 3 ingredients in a large bowl; add calamari. Cover and chill 2 hours or overnight. Sprinkle calamari with 1/2 teaspoon sea salt and black pepper.
Grill calamari over medium-high heat (350° to 400°) for 3 minutes on each side or until opaque. Cut tubes into rings; leave tentacles whole.
Combine potatoes, calamari, parsley, and 1/2 cup Red Wine Vinaigrette in a large bowl. Combine remaining vinaigrette and dandelion greens in a large bowl; toss to coat. Serve potato salad over dressed greens.