Heat two tablespoons of oil in a large Dutch oven or deep skillet over medium-high heat. Add the chopped onions and 1/2 teaspoon salt and cook, stirring frequently for 5 minutes. Add the chopped bell pepper and celery and continue to sauté, stirring frequently for another 5 minutes or so, until the vegetables look wilted.
While the onions, celery, and bell pepper are sautéing, place the peas in a colander and sort through, removing any discolored ones or debris. Rinse the peas with water and drain.
Add the minced garlic and jalapeño to the onion mixture and cook for 1-2 minutes, stirring often.
Add the drained peas, broth, black pepper, and cayenne pepper to the pot and raise the heat to high. Bring to a boil, stir, then lower the heat to simmer. Partially cover and continue to cook for 1 1/2 -2 hours, gently stirring occasionally, until the peas are close to being soft.
Meanwhile cut the smoked sausage links in half lengthwise and in half lengthwise again. Slice the lengths into 1/4- to 1/2-inch thick pieces.
Add the sausage pieces to the bean mixture and stir well. Continue cooking over low heat for 30 more minutes.
At the end of the cooking time, taste the bean mixture to see if it needs more salt or pepper.
Meanwhile, cook the rice.
Mix the cooked rice with the black-eyed pea mixture and allow to sit for 15 minutes. Garnish with chopped green onion tops and parsley, if desired. Serve, passing the Tabasco.