Set the potatoes and parsnips in a large pot. Cover with at least an inch of cold water, and add a generous pinch of salt. Bring to a boil, then lower heat to a simmer. Cook for 25-35 minutes, until very tender. Drain, then return to the pot (off the heat). Add 2/3 cup of milk, 2 tablespoons of the olive oil, and salt and pepper. Use a potato masher to mash until creamy, adding up to 1/3 cup more milk if needed. Cover and set aside.

Combine the lentils and 3 cups of water in a saucepan over medium-high heat. Bring to a boil, then reduce heat to simmer and cook until the lentils are tender and all the liquid has been absorbed (30-35 minutes). Remove from heat and set aside.

In a large skillet over medium heat, heat remaining 2 tablespoons of olive oil. Add the onions and garlic and cook until the onions are golden and translucent, 8-10 minutes. Add the carrots and celery, and cook until tender, about 8 minutes. Stir in the tomato paste and cook, stirring, for 2 minutes. Stir in the mushrooms and cook for 3 minutes. Pour in the red wine and bring to a simmer. Scrape any browned bits from the bottom of the pan, and cook, stirring fequently, until the liquid has reduced by half. Add the lentils, rosemary, thyme, and 1/2 cup of vegetable broth. Simmer for 3 minutes, stirring frequently, adding more broth if needed to keep the mixture moist but not liquid-y. Season to taste with salt and pepper.

Preheat oven to 350 F. Spread the lentil and vegetable mixture into a large baking dish in an even layer. Add the parsnip-potato mash and gently layer over the lentils. Bake 20 minutes, until the mash begins to brown. Serve hot.