With heavy-duty kitchen shears, cut out the backbones of both Cornish game hens. On a clean work surface, spread both of them open, skin up, and press down firmly with the heel of your hand to flatten them. Put the hens side by side in a clean nonreactive baking dish.

In a bowl, stir together the melted butter, red wine, lemon juice, thyme, garlic, shallot, and salt and pepper to taste. Spoon about a quarter of the mixture over the hens, turn them to coat them evenly, cover with plastic wrap, and marinate in the refrigerator for 2 hours. Reserve two thirds of the remaining sauce mixture for basting, and the final third for making a sauce.

Remove the hens from the refrigerator and leave them at room temperature while you preheat a grill to medium-low heat, about 300°F. Turn the hens in the sauce they’ve been marinating in, lightly oil the grill grate, and place them on the grill skin side down. Discard the portion of the sauce in which they marinated. Cover the grill and cook the hens for 17 minutes, basting them every 5 minutes or so with the sauce you’ve reserved for basting. Turn the heat up to medium and turn over the hens. Continue grilling, covered,  for 17 minutes longer, basting every 5 minutes, until the hens are evenly browned.

Remove the hens from the grill to a platter or cutting board, tent them with heavy-duty foil, and leave them to rest for about 5 minutes. Meanwhile, in a saucepan, warm the remaining sauce mixture. To serve, use a sharp, heavy knife to cut each hen in half down the center of its breastbone. Transfer to heated dinner plates and spoon the warm sauce over each portion. Garnish with parsley.