Combine chicken giblets, 1 cup water, and 1/2 teaspoon salt in a pressure cooker; cook for about 20 minutes.
Drain giblets and let cool, about 10 minutes. Chop giblets in a food processor or by hand; set aside.
Bring rice and water to a boil in a saucepan over high heat. Reduce heat to medium-low, cover, and simmer until the rice is tender, and the liquid has been absorbed, 20 to 25 minutes.
Spread cooked rice onto a baking sheet to cool, about 5 minutes. Transfer rice to a large bowl.
Mix in giblets, grated Parmesan cheese, marinara sauce, 1/4 cup bread crumbs, eggs, parsley, salt, and ground pepper. Cover with plastic wrap and refrigerate for 1 hour.
Remove rice and giblet mixture from refrigerator and form 2-inch, football shaped croquettes. Roll croquettes in bread crumbs and place on a baking sheet.
Heat oil in a large skillet over medium-high heat; cook breaded croquettes until browned on all sides, about 10 minutes. Transfer to a plate lined with paper towels to absorb excess oil.
