Preheat the oven to 375 degrees. In a large bowl, toss the potato halves or quarters with 1 tbsp truffle olive oil and a pinch of sea salt, taking care to coat each potato piece in the oil. Spread the potatoes on a large baking sheet covered with a sheet of parchment paper.
Bake the potatoes for 10 minutes, toss them with a wooden spoon and then bake them for about 10 minutes more or until fork tender. Remove the potatoes from the oven and carefully transfer them back to a large bowl.
Add the celery, shallot or red onion and green onion to the bowl with the potatoes.
In a small bowl, whisk together the vegan mayo, dijon mustard, pinch of sea salt and black pepper to taste. Add the mayo mixture to the bowl of potatoes and veggies and mix everything well, taking care to coat everything with the mayo and dijon mustard and seasonings. (This is easier to do while the potatoes are still warm.) Taste the mixture and adjust the seasonings. If desired, add an additional teaspoon of truffle oil to the mixture and toss well once more (adding more truffle oil will make for a stronger truffle flavor, so add carefully!)
Top the mixture with the chopped parsley and serve this roasted potato salad recipe immediately or store in the refrigerator until you're ready to serve it. Store leftovers covered in the refrigerator.