Heat a panini grill. Divide 1-ounce (1/3 cup) smoked Gouda between 1 top and 1 bottom of a croissant. Sprinkle each half with 1 teaspoon Parmesan. Top each side with about 3 slices of Genoa salami (about 1 ounce of salami per sandwich, total.) Top 1 half of the sandwich with a small handful of arugula. Close the sandwich and continue with the remaining sandwiches. Grill the panini until the cheese melts, about 3 to 4 minutes.
Cut each sandwich into thirds. Place the remaining arugula on a serving platter. Place the warm sandwich slices on the arugula. Serve hot.