Heat butter in a medium heavy pot over medium-high heat. Add onion and carrot and cook, stirring occasionally, until browned, about 6 minutes. Add squirrel, cider, Dijon mustard, tarragon, 2 parsley sprigs, and 1/2 tsp. each salt and pepper. Bring to a simmer and cover pot. Braise until meat is very tender, 2 1/2-3 hours. Let squirrel cool in braising liquid.
Once cool, remove squirrel from liquid and cut into serving pieces (hind legs, front legs, saddle). Strain braising liquid, reserving vegetables. Bring liquid to a boil and reduce, uncovered, stirring occasionally, until sauce measures 1/2 cup.
Preheat the broiler. Brush squirrel pieces with some of sauce. Broil until browned and heated through, 8-10 minutes.
Meanwhile, pick parsley leaves. Toss with shallot, oil, vinegar, and 1/2 tsp. each salt and pepper.
Serve squirrel with parsley salad, reserved vegetables, and remaining sauce.