Heat the olive oil in a large soup pan over medium high heat. Add the onions and sprinkle them with 1/2 teaspoon salt. Cook, stirring frequently, for 8-10 minutes, until the onions have begun to become golden around the edges. Add the celery root and carrots and continue cooking for another 5 minutes or so, stirring now and then.
Stir in the garlic and jalapeño and cook for one more minute.
Stir the tomato paste into the sautéed onion mixture. Cook, stirring constantly for about 2 minutes, to help remove the raw tomato taste.
Add the broth, stirring to combine it with the tomato paste-onion mixture, then add the cans of fire-roasted tomatoes along with their juices and the chopped potato. Stir to combine. Drop in the serrano chile.
Bring the soup to a boil over high heat, then reduce the heat to low. Simmer for about 45 minutes. All of the vegetables in the soup should be soft. If they’re not, cook it a little longer. Fish out the serrano chile.
Using a ladle, transfer the soup in batches to a blender. Process until the soup is puréed, transfer to a large bowl, and then do the next batch, always being careful as you handle the hot liquid. (Or use an immersion blender to purée the soup mixture right in the pot.)
If using the blender, return the soup to the cooking pan and stir in the parsley. Taste for salt and pepper, adding what is needed.