Cook parsnip in boiling water 12 minutes or until very tender; drain. Cool to room temperature.
Place corn in a food processor; pulse 3 to 4 times or until chopped. Lightly spoon flour into a dry measuring cup; level with a knife. Add parsnip, flour, and next 4 ingredients (flour through eggs). Pulse 3 to 4 times or until combined. Stir in chives.
Heat 1 teaspoon butter in a large nonstick skillet over medium-high heat. Drop batter by heaping tablespoonfuls to form 6 cakes. Cook 2 minutes on each side or until golden brown. Repeat procedure with remaining butter and batter.