Heat a large skillet over medium-high heat. Add half of the olive oil and heat until it shimmers. Add half of the zucchini slices so they lay in one single layer. Cook until deep golden brown, about 4 minutes. Sprinkle with 1/4 teaspoon salt. Flip each slice and cook an additional 4 minutes to brown the other side. Remove to a paper towel-lined plate and continue with the remaining oil, zucchini and salt. Place drained zucchini on a platter and sprinkle with the Parmesan.
When all the zucchini has been removed, turn off the heat and pour off all but 1 tablespoon of oil from the skillet. Add the anchovy paste and break apart with a wooden spoon until it dissolves and smells toasted. Add the capers, oregano, red pepper flakes and tomato. Squeeze in the lemon juice and stir. Spoon the tomato mixture over the zucchini and sprinkle with the chopped parsley.