Slice the potatoes 1/3-inch thick and put in a saucepan with 1 teaspoon of the salt, the water, and the wine. Tie the parsley stems, peppercorns, and thyme in a piece of cheesecloth and add to the pot. Bring to a boil over high heat and immediately lower the heat to maintain a gentle simmer. Cook until the potatoes are tender but not mushy, about 12 minutes. Drain; discard the herb bundle and transfer the potatoes to a bowl.
While the potatoes are cooking make the dressing. In a small bowl, whisk together the vinegar, parsley, shallots, the remaining 2 teaspoons of the salt and pepper. Gradually whisk in the oil to make a dressing. Toss the warm potatoes with the dressing gently so as not to break up the potatoes. Allow the salad to sit at room temperature for 1 hour to marinate, tossing several times to coat the potatoes. Season with salt and pepper, to taste. Serve at room temperature.
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