Bring 1 1/2 cup water to a boil in a saucepan; add oats. Reduce heat to low and simmer, stirring constantly, until oats have absorbed water, 1 to 3 minutes.
Heat 2 tablespoons olive oil in a skillet over medium heat; cook and stir onion, carrots, celery, and garlic in the hot oil until tender, about 10 minutes. Mix 2 tablespoons water, basil, and salt into the onion mixture. Cover skillet and cook, stirring occasionally until water is absorbed, 5 to 10 minutes.
Transfer onion mixture and oat mixture to a food processor; add adzuki beans and parsley. Puree until mixture is fully blended. Stir brown rice flour slowly into mixture until fully incorporated. Form mixture into 8 patties.
Heat 2 tablespoon olive oil in a skillet over medium heat; fry patties in the heated oil until lightly browned, about 5 minutes per side.