Preheat the oven to 450 degrees . In a medium bowl, whisk together 1/4 cup olive oil and the vinegar, then whisk in the finely chopped parsley and garlic. Stir in the celery and season with salt and pepper.
Season the pork with salt and pepper. In a large cast-iron skillet, heat 1 tablespoon olive oil over medium-high heat. Add the pork and cook until browned all over, about 5 minutes. Transfer the pan to the oven and roast until the pork registers 150 degrees on an instant-read thermometer, about 12 minutes. Let the pork rest for 10 minutes before thinly slicing.
Meanwhile, using a food processor, chop the whole garlic clove. Add the arugula and whole parsley leaves and pulse until finely chopped; scrape down the bowl. Add the pecans and cheese and season with salt and pepper; pulse until combined. With the machine on, slowly add the remaining 3 tablespoons olive oil. Serve the pesto with the pork and celery slaw.