In a large pot of boiling salted water, cook the rutabagas until crisp-tender, about 5 minutes. Using a slotted spoon, transfer the rutabagas to a large baking sheet and let cool. Add the parsley to the boiling water and cook until tender, about 4 minutes. Drain and let cool slightly. Squeeze dry and coarsely chop.
Melt the butter in a large skillet. Add the shallots, jalapeños and garlic. Season lightly with salt and pepper and cook over low heat, stirring occasionally, until softened, about 10 minutes. Add the stock and cook over moderately high heat until reduced by half, about 5 minutes. Add the cream and simmer over low heat until reduced by one-third, about 8 minutes. Stir in the parsley and simmer until thickened, about 4 minutes. Season generously with salt and pepper.
Preheat the oven to 400°. Butter a 10-by-15-inch glass baking dish. Arrange half of the rutabaga slices in the dish. Season with salt and pepper. Using a slotted spoon, cover the rutabagas with the parsley and pour half of the cream over the parsley. Top with the remaining rutabagas and pour the remaining cream over them; press gently to even out the slices. Cover with foil and bake for about 1 hour, or until bubbling.
Position an oven rack 6 inches from the heat and preheat the broiler. Sprinkle the Parmesan cheese over the rutabagas and broil, rotating the dish as necessary, until browned, about 2 minutes. Let cool for at least 15 minutes before serving.
Make Ahead: The recipe can be assembled and refrigerated overnight. Bring to room temperature before baking.