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Place potatoes, 1 tablespoon salt, ½ tablespoon vinegar, and 3 cups tepid water in a medium saucepan. Bring to a boil over high heat, stirring occasionally until salt is dissolved. Reduce to a bare simmer and cook until potatoes are completely tender and show no resistance when poked with a paring knife or cake tester, about 17 minutes. Drain potatoes and allow to cool to warm room temperature, about 10 minutes.
Meanwhile, add onion to a medium bowl and add just enough vinegar to cover. Let stand until onion is tender and lightly pickled, about 20 minutes.
Using a blender, immersion blender, or food processor, blend tarragon, parsley, garlic, and oil until well pureed. Mix in espelette pepper and season with salt and pepper.
Drain onions. In a serving bowl, toss potatoes with tarragon pesto, onions, and olives, and serve.