Melt 2 tablespoons of the tallow over high heat in the pressure cooker, Sprinkle the oxtails liberally with salt and pepper, then brown in the fat, working in batches if necessary. Transfer the browned oxtails to a plate and set aside.
Reduce the heat to medium and add the remaining tablespoon of tallow, if necessary. Cook the onions, stirring frequently, until the onions are soft and golden, about 5 minutes. Add the garlic and cook one more minute.
Return the browned oxtails to the cooker and stir in the celery, carrots, wine, beef stock, tomato paste, mustard, and bay leaf. Tie together the thyme and parsley with kitchen twine and add that to the oxtails, vegetables and liquid in the cooker.
Lock the lid of the pressure cooker in place and increase the heat to high until the cooker reaches full pressure (15 psi). Reduce the heat to medium, and cook for 55 minutes. Remove from the heat and allow the pressure to decrease naturally.
Carefully unlock and remove the lid of the pressure cooker. Remove the tied herbs and bay leaf; skim any excess fat from the surface of the stew. Remove the oxtails and transfer to a plate; shred the meat away from the bones with a fork and return to the pressure cooker. Taste and season as necessary with salt and pepper. Serve over mashed potatoes or parsnips, and garnish with sliced green onion, if desired.