Slice mushroom caps in half and place in a wide-rimmed bowl. Combine oil, paprika, garlic powder, cumin, and black pepper in a small bowl. Brush mushrooms with oil-spice mixture, and let sit 15 minutes.
Next, heat a greased grill or grill pan over medium-high heat. Add mushrooms (in batches, if necessary, to avoid over-crowding). Cook 3 to 4 minutes, undisturbed, until grill marks appear, flip and cook 2 more minutes. Season with 1/2 tsp. of the salt, transfer to a cutting board, and slice into strips.
Next, add green onions in a single layer to pan. Cook over medium-high until charred and fragrant, about 3 minutes, toss and cook 1 to 2 minutes, until limp; transfer to a cutting board.
Mash avocados in a bowl with 1/4 tsp. salt, lime zest, and juice; set aside.
Once green onions are cool enough to handle, finely chop and place in a bowl. Add parsley, cilantro, lemon zest and juice, and olive oil; mix well. Season to taste with salt and pepper.
To assemble tacos, spread avocado evenly on tortillas and top with mushrooms and red cabbage. Lastly, spoon scallion salsa verde over top and serve.