Preheat a grill or grill pan over medium-high heat.
For the pork sliders: Combine the pork, parsley, scallion, garlic, 1 tablespoon salt and 1 teaspoon pepper in a medium bowl. Mix well and divide into 4 equal patties. Pat the patties dry with a paper towel and season both sides with more salt.
Grill the patties until cooked through and nice grill marks develop, about 4 minutes per side. Transfer to a platter to rest.
For the grilled peach salsa: Lightly coat the cut sides of the peach halves with 1 tablespoon of the oil.
Grill the peaches cut-side down until slightly softened and nice grill marks develop, 4 to 5 minutes per side. Transfer to a cutting board until cool enough to handle, then dice. Whisk together the balsamic vinegar, sugar, shallot, remaining 1 tablespoon oil and a pinch of salt in a large bowl. Tear the mint leaves into the bowl. Add the diced peaches and gently stir to combine.
Place the pork patties on the buns and top with the grilled peach salsa.