Peel the plantains and cut into diagonal slices ¼ to ⅓ inch thick.
Pour oil into a large skillet to a depth of about ¼ inch and heat until the surface of the oil shimmers.
Add the plantain slices, a handful at a time (do not crowd skillet), and fry until nicely caramelized, adding oil if necessary.
Drain on paper towels and keep warm.
In a large, heavy saucepan, heat 1 tablespoon oil over medium heat, add rice, and sauté until the grains turn opaque.
Pour enough broth into the skillet to cover the rice by about 1 inch (if broth is low-sodium, add salt to taste).
Cook over medium-high to high heat, uncovered, until the liquid has been absorbed and steam holes appear in the rice.
Turn off heat, cover, and let stand for 10 minutes.
Uncover and fluff with a fork.
To serve, put the rice in a large bowl and top attractively with plantain slices; garnish with parsley.