Place the saffron threads into a small cup or bowl and cover them with the warm water. Let them soak for at least 10 minutes. You can begin cooking in the meantime.
Coat the bottom of a large 15-inch paella pan (Note 2) with 3 tablespoons of the olive oil and place it over medium heat. When the oil is hot, add the mushrooms. Season them with a few pinches of salt. Cook the mushrooms for about 10 minutes, flipping once halfway through, until lightly browned on both sides. Remove the mushrooms from the pan and transfer them to a plate.
Add the remaining tablespoon of oil to the pan. When the oil is hot, add the onion and bell pepper. Sweat the onion and pepper for about 5 minutes, stirring often, until they begin to soften.
Push the onion and pepper to the side, then add the garlic in the space you've just made. Sauté the garlic for about 1 minute, stirring constantly, until it becomes very fragrant.
Stir in the tomatoes, wine, sweet paprika, smoked paprika, and the saffron, along with its soaking water. Raise the heat and bring the liquid to a boil. Lower the heat and allow the mixture to simmer until the liquid reduces by about half, about 5 minutes.
Return the mushrooms to the pan and stir in the rice, broth, and olives. Season the mixture with a bit more salt and some black pepper. Raise the heat again and bring the liquid to a boil, stirring occasionally as it heats up. Once the liquid boils, lower the heat so that it's cooking at a low simmer. Stop stirring at this point. Let the mixture simmer, uncovered for about 20 minutes, until the rice is tender and slightly charred at the bottom where it contacts the pan. Reposition and wiggle the pan around occasionally so that the paella cooks evenly (Note 3). If the mixture gets too dry before the rice is tender, you can add up to a half cup of water at a time.
Sprinkle the peas evenly over the paella, and let it cook for a minute or two longer, just until the peas are warmed through.
Remove the pan from heat. Cover it with a lid or foil and let it sit for 5 minutes.
Uncover the paella. Season it with some salt and black pepper to taste, then sprinkle it with fresh parsley. Serve with lemon wedges.