Preheat oven to 425 degrees F with racks in the upper and lower third. Cut four large rectangular pieces of parchment paper, about 12 x 16 inches. Fold each in half crosswise, then unfold and lay flat.
Toss mushrooms and spring onion with oil in a bowl. Season with salt and pepper.
Pat cod fillets dry, and arrange each on individual pieces of parchment, on one side of the crease. Season both sides of the fish with salt and pepper. Spoon over shiitake mixture, dividing evenly, and drizzle each with 1 tablespoon of wine. Fold each piece of parchment over and overlap small pleats to seal and create four half-moon–shaped packets.
Divide packets between two baking sheets and bake in the upper and lower third of the oven until the fish is just cooked through, 17 minutes.
Meanwhile, make the pea relish: In a small pot of boiling water, boil peas until just tender, about 3 to 4 minutes. If using frozen peas, boil to reheat, about 1 minute. Drain and transfer to a medium bowl.
Using the back of a fork or a potato masher, gently crush peas. Stir in the lemon zest, juice, and mustard and season with salt and pepper.
Once fish is cooked, carefully cut open parchment packets to release steam. Spoon pea relish over each piece of fish and sprinkle with mint leaves. Serve with lemon wedges and a generous pinch of Aleppo pepper, if using.