Wash the watermelon radishes to remove any dirt and trim off the ends. Using a mandoline or a sharp knife, thinly slice the radish into paper thin discs.
Layer the radish, tomatoes, and melon on a plate and sprinkle with about one teaspoon of vinegar and two teaspoons of extra-virgin olive oil. Taste for balance, the vinegar should be assertive but not harsh.
Once the dressing is where you want it, crumble the chèvre on top and garnish with the chopped parsley.
Season with a generous pinch of flakey sea salt and freshly cracked black pepper.