Preheat oven to 400°F.
Using a mandolin if possible, thinly slice the parsnips crosswise into rounds. Place them in a bowl, add the olive oil and salt, and toss. Arrange in a single layer on a baking sheet and bake until golden and crisp, turning once, about 20 minutes. Remove to paper towels to drain. Sprinkle with the sea salt.
Coarsely chop the figs and put them in a sauce- pan with the water. Over medium heat, bring to a simmer. Continue to cook, stirring, until the water is absorbed and the figs are soft, about 5 minutes. Remove to a food processor or blender and add the olives, olive oil, balsamic vinegar and the capers. Coarsely purée. Taste and adjust seasonings as desired.
Serve the warm chips accompanied by the spread.