Boil water, sugar, garlic, peppercorns, and 3 tablespoons salt in a 4-quart pot, uncovered, 10 minutes.
While brine is boiling, halve cucumbers lengthwise and cut crosswise into 1/3-inch-thick slices.
Remove brine from heat and add cucumbers and radishes. Let stand, uncovered, 10 minutes, then drain in a colander, discarding garlic and peppercorns. Transfer cucumbers and radishes to a bowl of ice and cold water to cool, then drain well in colander.
Toss cucumbers and radishes with parsley, oil, and remaining 1/2 teaspoon salt in a large bowl and chill, uncovered, about 20 minutes.
