Fill a large pot with water; add vinegar. Bring to a boil. Add potatoes and simmer until tender, about 15 minutes. Drain.

Preheat oven to 300 degrees F (150 degrees C). Coat 2 rimmed baking sheets with 2 tablespoons olive oil each.

Arrange potato slices on a towel. Dab with another towel to dry. Arrange potato slices on the oiled baking sheets.

Bake in the preheated oven until bottoms are golden brown, about 20 minutes. Flip all slices. Continue baking until tops are golden brown, about 10 minutes. Remove from oven.

Increase oven temperature to 350 degrees F (180 degrees C). Coat a 9x12-inch baking dish with the remaining 2 tablespoons oil.

Combine ham, cream of mushroom soup, onion, milk, capers, salt, cayenne pepper, and chili powder in a large bowl.

Spread ham mixture in the bottom of the baking dish. Cover evenly with Cheddar cheese and pepper Jack cheese. Arrange potato slices on top. Cover potato slices evenly with mozzarella cheese.

Bake in the preheated oven until mozzarella cheese is melted, about 15 minutes. Garnish with parsley.