In a 2 quart stock pot, combine water, wine, white vinegar, 2 lemons, 2 TB salt, peppercorns and the sprigs of parsley and thyme. Bring to a boil, reduce to a simmer and add the calamari. Leave in the water for 3-4 minutes to partially cook the calamari. Strain and spread out on a pan to cool completely. Discard the poaching liquid.
Stack the basil leaves, roll them tight, and slice thin with a sharp knife (chiffonade)
Once the calamari has cooled, combine with the juice of the remaining 2 lemons, red wine vinegar, tarragon vinegar, oil, 1 tsp salt, black pepper, garlic, parsley leaves and basil. Cover and marinate in the refrigerator for a couple hours. Serve chilled.