Lightly mix the turkey, grapes, Parmesan cheese, and mayonnaise in a bowl until well combined. Season to taste with salt and black pepper.
Place the 4 tortillas out onto a work surface, and place 1 spinach leaf in the middle of each. Spoon 1/4 of the turkey filling into each spinach leaf, in a line. Fold up the bottom of each tortilla to hold the filling, then wrap tightly. Leave the tops open.