Combine the black-eyed peas, 3 cups water and a pinch of salt in a small pot. Cover and bring to a boil.
Meanwhile, heat the olive oil in a large, deep skillet over medium heat. Add the kielbasa and cook until golden, about 2 minutes. Add the scallion whites, celery, garlic, jalapeno, jerk seasoning and a generous pinch of salt. Cook until the vegetables brown, about 5 minutes. Add the thyme and tomato paste and cook, stirring occasionally, until the tomato paste turns brick red, about 2 more minutes. Add the rice, bay leaves and the black-eyed peas with their liquid to the skillet and bring to a boil; do not stir.
Add the collards; cover, reduce the heat to low and simmer undisturbed until most of the liquid is absorbed, about 50 minutes; set aside for 10 minutes. Remove and discard the bay leaves.
Just before serving, add the scallion greens and fluff with a fork.