Cook the lobsters:
With your hands, pull the tail sections and whole claws off the lobsters’ bodies. Set aside the bodies. In a large stockpot, bring 1 gallon of the water and the vinegar to a rolling boil. Add the tails and claws. After 2 minutes, remove the tails. After 5 additional minutes, remove the claws.
Remove the meat:
While the lobsters are still warm, remove the meat from their shells. Using poultry shears, slit the underside of each tail shell and pull out the meat, then, using a chef’s knife, halve the meat lengthwise and remove the center vein from both sides. Place the claws between two kitchen towels and whack them with the back of a sturdy knife. Then use the shears to cut through the shells lengthwise, far enough so you can pop out the claw meat. Remove the cartilage from the centers of the claws and then dice the claw meat into 1/2-inch pieces. Reserve all shells for the sauce. Cover the tail and claw meat separately and refrigerate both.
Make the lobster stock:
Use poultry shears to split the reserved lobster bodies in half lengthwise. Inside the head on both sides, you will see feathery gills. Cut away and discard them. Use a cleaver or chef’s knife to chop the bodies into approximately 2-inch pieces. In a large, heavy saucepan over medium heat, melt the 1/4 cup butter. Stir in the celery, shallots, fennel, and carrots and let them sweat until soft but not at all brown, about 5 minutes. Stir in the lobster shells (including the tail and claw shells) and tomato paste. Add enough water to cover the shells by 1 inch. Increase the heat to medium-high and bring the liquid to a simmer. Lower the heat to medium and maintain the simmer for 45 minutes. Throughout the simmering, skim and discard any foam or impurities that rise to the top. Add the thyme, bay leaf, and tarragon and cook for another 10 minutes. Strain the stock through a large-mesh sieve into a large container. Clean the pot and strain the stock into it through a fine-mesh strainer. Simmer the stock over medium-high heat until it is reduced to 1/2 cup, about 90 minutes.
Meanwhile, blanch the parsnips:
Put the parsnips, salt, and sugar in a heavy saucepan, cover with cold water, and bring to a boil over high heat. Lower the heat to medium and simmer until the parsnips are tender but still firm, about 5 minutes. Transfer the pan to the sink and run cold water into it in a thin stream for about 6 minutes to slowly stop the cooking process and cool the parsnips completely. Drain the parsnips. If more than a few minutes remain to complete the stock, refrigerate the parsnips.
Finish the parsnips:
In a slope-sided sauté pan over medium heat, melt 1 tablespoon of the 13 tablespoons of butter. Stir in the ginger and cook until tender, about 3 minutes. Add the blanched parsnips and lobster stock. Increase the heat to medium-high and bring to a boil. Stir in the remaining 12 tablespoons of butter. Once it is incorporated, add the claw meat, lime juice, and cilantro. Keep warm over very low heat.
Poach the lobster tails:
Run a bamboo skewer lengthwise through each tail half to keep them from curling when poaching. In a medium saucepan over medium-high heat, bring the remaining 3 tablespoons of water to a boil. Lower the heat to medium and whisk in the cubes of the pound of butter one by one until completely incorporated. Then decrease the heat to low, add the reserved lobster tails, and poach them until a cake tester inserted into the center of one and pressed to your lips feels warm, about 8 minutes. (Be careful not to overcook the lobster or it will be tough.)
Present the dish:
Divide the parsnip mixture among 4 pasta bowls and top each with 2 lobster tail halves. Garnish with micro cilantro and serve immediately.