Cover the noodles with hot tap water in a large bowl and soak, flipping once, for 7 minutes; they should pull apart without breaking. Drain in a strainer and shake to dry; pat with towels if needed. Return to the bowl and toss with 1 tablespoon olive oil and a pinch of salt.
Add the chard and parsley to the bowl; toss, pulling apart the noodles. In a small bowl, beat the eggs with the parmesan cheese and salt and pepper to taste; add to the noodles and toss to coat.
Heat the remaining 2 tablespoons oil in a 9-inch skillet with a lid over medium heat. Add the noodle mixture and press down with a spatula; cover and cook until the bottom is golden brown, about 8 minutes. Place a large plate over the pan; invert the cake onto the plate. Slide the cake back into the pan, cover and cook until golden on the other side, 5 to 10 minutes. Invert onto a plate, sprinkle with salt and cut into wedges.