Remove giblets and neck from capon; reserve for other uses. Rinse capon thoroughly with cold water; pat dry. Sprinkle 1 teaspoon salt over surface and in cavity of capon.
Melt butter in a large skillet; sauté onion and celery until transparent. Remove from heat; stir in next 9 ingredients, mixing well. Stir oysters into stuffing mixture.
Place stuffing in cavity of capon; close with skewers. Tie ends of legs to tail with string or tuck them under flap of skin around tail. Lift wingtips up and over back, tucking under bird securely.
Brush entire bird with oil; place breast side up in a roasting pan. Insert meat thermometer in breast or meaty part of thigh, making sure it does not touch bone. Bake, covered, at 325° for 3 hours. Remove cover, and bake an additional 30 minutes until meat thermometer registers 185° or drumsticks are easy to move.
Transfer capon to serving platter. Let stand 15 minutes before carving. Garnish with spiced peaches and parsley, if desired.