1. Place the potatoes in a large pot and add water to cover. Heat to a boil, then lower the heat and simmer uncovered until fork-tender, about 25 minutes. Drain in a colander and let cool a few minutes.

2. While the potatoes are cooking, make the Caesar dressing. Place the anchovies, garlic, mustard, thyme, egg yolk, vinegar, and lemon juice in a food processor and process just to combine. With the machine running, pour the oils through the feed tube in a thin, steady stream to make a thick mayonnaise. Add the sun-dried tomatoes and pulse just to combine. Season the dressing to taste with salt and pepper.

3. Cut the warm potatoes into large uneven chunks. Toss with a generous amount of Caesar dressing to bind, then mix in the parsley and Parmesan. Serve at once or at room temperature.