Preheat the oven to 350 degrees F.
Heat a splash of vegetable oil in a large skillet. In batches, cook the eggplant slices until beginning to brown slightly, 1 to 2 minutes per side. Transfer to a paper-towel-lined baking sheet; let cool.
Mix the cheese and parsley in a small bowl. Smear a dollop of the mixture onto each eggplant slice, top with a strip of ham, and roll up the eggplant to enclose the filling. Use a toothpick to keep the rolls closed if necessary.
Transfer the rolls to a casserole dish. Bake until the eggplant is tender and the cheese begins to ooze out, 15 minutes, then broil for 1 or 2 minutes to crisp the edges.
Serve over a bed of arugula.