Preheat the oven to 400 degrees F. Cut the squashes in half, down the equator. Scoop out the seeds and discard.Remove the pith from 1 quarter of preserved lemon, and mince the rind.
Place the squash halves, cut side up, on a rimmed baking sheet and drizzle the flesh with a few tablespoons of olive oil, salt, pepper, diced preserved lemon rind, and a squeeze of lemon. Roast uncovered in the oven for 1 hour.
Remove the squash from the oven. Cut into large wedges, drizzle with another few tablespoons of olive oil and lemon juice, salt, pepper, and chopped parsley.
Bring a large pot of salted water to a rolling boil. Add the pasta to the boiling water. Boil the pasta until it's al dente.
Cut the skin from the roasted squash wedges and discard. Chop the squash flesh into 1/2-inch chunks.Remove the white pith from the remaining 2 quarters of preserved lemons and discard. Dice the yellow rind.
Once the pasta is cooked, but still has a little bite, remove with a slotted spoon to a large platter. Drizzle with the remaining olive oil, add the remaining preserved lemon, squash, grated Pecorino Romano, and a splash of the warm pasta water, to create a soupy sauce in the bottom of the platter. Add black pepper and salt, to taste, and chopped parsley, for garnish. Serve immediately.