Garlic-Aioli Roasted Turkey with Lemon-Parsley Pan Sauce

Pat turkey dry, reserving giblets and neck. Let turkey stand at room temperature 1 hour to air-dry the skin before roasting.
Meanwhile, position rack in lower third of oven and preheat to 450°F. Trim 1/4" from top of 3 garlic heads to expose cloves, then place heads on a sheet of foil. Spoon oil over garlic, then wrap tightly in foil to enclose. Transfer to a rimmed baking sheet and roast until very soft, about 40 minutes.
Carefully unwrap garlic, let cool slightly, and squeeze pulp into a food processor. Add mayonnaise and blend until smooth.
Place turkey on rack in large roasting pan. Cut remaining 2 garlic heads in half crosswise and stuff cavity of bird with garlic head halves, bay leaves, parsley, and thyme. Loosely tie legs together with kitchen twine. Starting at neck end, slide your hand between skin and breast meat to loosen skin. Rub half of mayonnaise mixture under skin, then half over outside. Season both sides with sea salt and pepper. Turn turkey breast side up and place reserved giblets and neck in pan.
Place turkey in 450°F oven, then reduce oven temperature to 350°F and roast 1 hour. Rotate pan, wrap wing tips in foil, and continue roasting until an instant-read thermometer inserted into the thickest part of the thigh registers 160°F, 1–1 1/2 hours more.
Transfer turkey to a cutting board, reserving turkey drippings and juices in pan, and cover loosely with foil. Let rest at room temperature 30 minutes before carving (internal temperature will rise to 165°F and juices will lock in).
Transfer pan drippings and juices to a 2-qt. glass measuring cup, then skim off fat, reserving 2 Tbsp. fat. Add enough stock to pan juices to make 4 1/2 cups total. Combine cornstarch and reserved fat with a fork in a small bowl.
Set roasting pan across 2 burners. Remove garlic from interior of turkey, then add to pan along with shallot, bay leaf, and thyme. Cook over medium-high heat, stirring occasionally, until shallots begin to brown, about 5 minutes. Deglaze pan with pan juices mixture, stirring and scraping up brown bits, about 2 minutes. Whisk in cornstarch mixture and cook until liquid is reduced by one-third, about 5 minutes. Strain through a fine-mesh sieve into a large bowl and stir in parsley, lemon juice, kosher salt, and pepper.