Combine the first 5 ingredients in a saucepan; bring to a boil, stirring until salt dissolves. Remove from heat; cool completely. Remove and discard giblets and neck from chicken. Rinse chicken with cold water; pat dry. Trim excess fat. Pour brine into a 2-gallon zip-top plastic bag. Add chicken; seal. Refrigerate 8 hours or overnight, turning the bag occasionally.
Preheat oven to 400°.
Bring 2 cups cider to a boil in a small saucepan over medium-high heat. Cook until cider has thickened and reduced to 1/4 cup (about 15 minutes). Set aside.
Remove chicken from bag; discard brine. Pat chicken dry with paper towels. Place the onion halves, parsley, and garlic into cavity. Lift wing tips up and over back; tuck under chicken. Tie legs. Place chicken on rack of a broiler pan. Bake at 400° for 1 hour and 30 minutes or until thermometer registers 175°. Remove from oven (do not turn oven off). Carefully remove and discard skin. Baste chicken with half of reduced cider; return to 400° oven for 10 minutes. Remove from oven; baste with remaining cider reduction. Transfer chicken to a platter.
Place a zip-top plastic bag inside a 2-cup glass measure. Pour drippings into bag; let stand 10 minutes (fat will rise to the top). Seal bag; carefully snip off 1 bottom corner of bag. Drain drippings into a small bowl, stopping before fat layer reaches opening; discard fat. Serve jus over chicken.