Heat the oil in a large skillet over medium-high heat.
Add the sausages and cook, turning occasionally, until browned and cooked through, 10 to 12 minutes.
Meanwhile, in a food processor fitted with the coarse grating disk, grate the celery root.
In a large bowl, whisk together the sour cream, lemon juice, mustard, ½ teaspoon salt, and ¼ teaspoon pepper.
Add the celery root, onion, parsley, and pickles and toss to combine. Serve with the sausages.