In medium saucepan, bring 3 cups of water to a boil. Reduce the heat to medium low and add lobster tails, salt and Old Bay seasoning. Simmer for 8-10 minutes, until the shell turns pink and the meat is opaque.
Take the lobster tails out of the water and remove the meat. Finely chop the lobster meat.
Put the shells back into the water. Add garlic cloves, bay leaf and thyme. Increase the heat to medium high and boil until the broth is reduced to about ½ cup, about 25 minutes.
Meanwhile, bring another pot of water to a boil. Place the eggs in a steamer and steam over boiling water for 15 minutes. Then submerge the steamed eggs in cold water until cooled. Then peel the eggs, cut in half and separate the egg yolks.
In a medium bowl, mash the egg yolks with a fork and stir in mayo, parsley and hot sauce. Mix in chopped lobster. Then add about 3 tablespoons of lobster broth into the mixture a tablespoon at a time until desired thickness reached. Transfer the filling into a piping bag, or a zip lock bag with one end cut, and pipe the filling into the egg whites. Garnish with chopped parsley, if desired. Serve immediately.