Lay one flour tortilla flat on a clean surface.
In a medium bowl, combine shredded chicken, chopped romaine lettuce, Caesar dressing, shredded Parmesan cheese, sun-dried tomatoes, and chopped parsley. Mix well until all ingredients are evenly coated.
Season the mixture with salt and freshly ground black pepper to taste.
Spread half of the chicken Caesar mixture evenly over the tortilla, leaving about a 1/2-inch border around the edges.
Starting from one edge, carefully roll the tortilla tightly into a log shape.
Repeat steps 2 to 5 with the second tortilla and remaining chicken Caesar mixture.
Wrap each rolled tortilla log tightly in plastic wrap and refrigerate for at least 10 minutes to firm up.
Remove the plastic wrap and slice each roll into 1-inch thick pinwheels.
Arrange the pinwheels on a serving platter and serve immediately.