Preheat oven to 450°. Toss potatoes and carrots in bottom of large baking
dish with 1 tablespoon oil.Season with salt and pepper.
Pat chicken dry with paper towels, drizzle with remaining tablespoon oil,
and season with salt and pepper. Arrange, skin-side up, on top of
vegetables in baking dish.
In a small bowl, stir together mustards, maple syrup, thyme, and
red pepper flakes. Brush on top of chicken and drizzle any extra over
vegetables. Roast chicken until golden and a thermometer inserted into
thickest part of thigh registers 160°, about 35 minutes.
Remove from oven and brush thighs with chicken drippings.
Turn oven to broil and broil thighs until deeply golden, 2 to 3 minutes.
Let cool slightly, then sprinkle with parsley before serving.