Cut parchment paper into 11-inch circle, then cut 1-inch hole in center, folding paper as needed.
Bring water, oil, and salt to simmer in 12-inch skillet over high heat. Off heat, add carrots, top with parchment, cover skillet, and let sit for 20 minutes.
Uncover, leaving parchment in place, and bring to simmer over high heat. Reduce heat to medium-low and cook until most of water has evaporated and carrots are very tender, about 45 minutes.
Discard parchment, increase heat to medium-high, and cook carrots, shaking skillet often, until lightly glazed and no water remains, 2 to 4 minutes. Transfer carrots to serving platter and serve.
Microwave figs and water in bowl until hot, about 1 minute; let sit for 5 minutes. Stir in chopped green olives, minced shallot, extra-virgin olive oil, red wine vinegar, parsley, ground fennel, and kosher salt. Spoon relish over carrots before serving.