Preheat oven to 475°. Line two or three large rimmed baking sheets with parchment paper. Trim carrot tops to 1 in. Peel carrots and halve any that are wider than 1/2 in. Peel potatoes and cut into 1/2-in.-wide pieces. Slice bulbous base from butternut squash, then halve neck lengthwise. Peel one of the halves, then cut into 1/2-in.-wide pieces. Save remaining squash for another use (such as soup). Trim fibrous green stalks from fennel bulbs and cut bulb lengthwise into 1/2-in.-thick slices.
Arrange each vegetable on a kitchen towel, top with another kitchen towel, and roll up to dry. Working with one vegetable at a time, unwrap onto a baking sheet in a pile. Toss with 2 tsp. oil and a pinch or two of salt, then arrange in a single layer on sheet. Repeat with the remaining vegetables.
Roast vegetables, shaking once halfway through to help cook evenly, until just tender-crisp and lightly browned, 6 to 10 minutes.
Arrange vegetables and dips on a very large platter or several small platters. Sprinkle carrots with thyme, and sprinkle fennel with parsley. Serve at room temperature.
Make ahead: Through Step 2, up to 1 day, chilled and covered (add a couple of minutes to roasting time). Through Step 4, up to 2 hours.