Preheat the oven to 375˚. Combine the carrots, leeks and 1 cup water in a medium saucepan; season with salt. Bring to a boil, then reduce to a simmer and cook, stirring, until the carrots are tender, about 5 minutes (add a few tablespoons of water if necessary). Add the lentils and cook, tossing, until hot, 2 to 3 minutes. Remove from the heat. Add 2 tablespoons each vinegar and olive oil; toss. Season with salt and pepper. Set aside.
Meanwhile, season the cod all over with salt and pepper. Combine the mustard and garlic in a bowl; spread on one side of the fish. Combine the breadcrumbs and parsley in a shallow bowl. Press the mustard side of the fish into the breadcrumb mixture to coat.
Heat 2 tablespoons olive oil in a large ovenproof nonstick skillet over medium-high heat. Add the cod crumb-side down; cook until golden brown, about 3 minutes (rotate the skillet as needed for even browning). Carefully flip the fish, transfer the skillet to the oven and bake until cooked through, 8 to 10 minutes.
Meanwhile, toss the arugula in a large bowl with the remaining 1 teaspoon each vinegar and olive oil; season with salt and pepper. Divide among plates; top with the lentil mixture. Serve with the cod.