Cut the parsnips crosswise into 1 1/2-inch lengths. Slice each piece lengthwise 1/4 inch thick. Stack a few together, and trim the edges to make squares. Repeat with remaining pieces; set aside.
In a large bowl, combine olive oil, the vinegar, 1/4 teaspoon salt, and the pepper. Mix together with a fork. Taste, and adjust for seasoning. Add the mache and prosciutto, and toss to combine. Divide among four salad plates.
Meanwhile, in a medium saucepan over medium heat, heat grape-seed oil. Working in batches so as not to crowd the pan, add the parsnips, and cook until golden, about 1 minute per side. Transfer to paper towels to drain briefly.
Divide the hot chips among the salads, tucking some of the chips into the greens. Serve immediately.