Using the large holes of a box grater or the grater blade of your food processor, grate the beets.
(It’s not necessary to peel them first.)
In a large, lidded sauté́ pan, heat 2 tablespoons of the oil over medium heat. Add the onion and cook until it softens and begins to look translucent, 6 to 8 minutes. Add the beets and salt and toss to combine. Cover and cook for 10 to 12 minutes, until the beets are completely softened. Add the vinegar, toss to combine, and scrape up the browned bits from the pan with a wooden spoon. Set aside to cool slightly.
In a mixing bowl, coarsely mash the beans with a potato masher or fork. Fold in the beet mixture, the rice, parsley, and black pepper to taste. Adjust seasonings. Shape into six patties, flattening to a ½-inch thickness.
In a sauté́ pan, heat the remaining 2 tablespoons oil over high heat. Add the patties, in batches if necessary to avoid crowding. They should sizzle—this creates a nice crust. Cook for 1 minute. Reduce the heat to medium and cook for 2 or 3 minutes. Carefully flip the burgers and cook until browned and firm, 4 to 5 minutes longer.