In 2-quart saucepan, place eggs in single layer. Add cold water to at least 1 inch above eggs. Cover and heat to boiling; turn off heat. If necessary, remove pan from heat to prevent further boiling. Cover and let stand 20 minutes. Immediately run cold water over eggs or place them in ice water until completely cooled.
To remove egg shells, crackle shells by tapping gently all over; roll between hands to loosen. Peel, starting at large end. Cut lengthwise in half. (For a different twist, eggs can be cut crosswise in half. Cut a thin slice off the bottom of each egg so it will stand up without falling over.) Transfer yolks to small bowl; mash with fork. Set egg whites aside.
Stir mayonnaise, parsley, horseradish, mustard, salt and pepper into yolks. Fill whites with egg yolk mixture, heaping it lightly. (Use a teaspoon to fill the egg whites and to spread the yolk mixture easily and neatly into the whites.) Cover and refrigerate up to 24 hours.