In a blender or food processor, puree the tomatoes with the onion, garlic, and salt until smooth. Pass the puree through a strainer into a large liquid measuring cup; note the amount and reserve. Pour enough chicken broth into another liquid measuring cup to make 4 cups liquid total—you want to keep the two liquids separate, since you will add the puree ﬁrst.
Heat the oil in a medium saucepan over medium-high heat until hot but not smoking. Add the rice and cook, stirring often, until the rice becomes milky white, 3 to 4 minutes. Pour in the tomato puree, mix gently, and cook until the puree darkens, thickens, and has mostly been absorbed by the rice, about 3 minutes.
Stir in the chicken broth and add the parsley and pickled jalapeños, as well as the carrots, peas, and corn, if using. Cook for about 15 minutes, or until most of the liquid has been absorbed but there is still some moisture in the pan. The rice should be cooked and tender; if it is not but all the liquid has been absorbed, add the 2 tablespoons of water, cover again, and cook for 2 more minutes. Remove from the heat and let the rice rest, covered, for at least 5 minutes.
Fluff the rice with a fork and serve.