Preheat the oven to 375 degrees F. Rinse the chicken with cool water and pat dry. Place the chicken in a roasting pan on a rack, breast side up. Sprinkle the cavity of the chicken with salt and pepper. Tuck the wing tips under the back of the chicken.
Combine 4 tablespoons butter, parsley if using, 1/2 teaspoon salt and 1/4 teaspoon pepper in a small bowl. Loosen the skin on the breast of the chicken by gently wriggling fingers under the skin. Rub outside of the chicken with about 1 tablespoon of the butter mixture. Rub the remaining mixture under the skin of the chicken breast. Sprinkle chicken with additional salt and pepper. Roast the chicken for 1 hour 15 minutes.
Meanwhile, melt the remaining 1 tablespoon butter in a small saucepan over medium-low heat. Add the garlic and cook, stirring, until fragrant but not browned, about 2 minutes. Spoon the garlic out of the saucepan and discard. Add the apricot preserves, mustard and vinegar and cook, stirring, until warmed through and combined, about 2 minutes. Set aside to cool until ready to glaze the chicken.
Increase the oven temperature to 425 degrees F. Remove the chicken from the oven and brush evenly with the apricot glaze. Continue roasting until an instant-read thermometer stuck into the chicken thigh registers 160 degrees F, about 15 minutes more. Tip the chicken so the juices run into the pan. Transfer the chicken to a cutting board and loosely tent with foil.
Pour the juices from the pan into a degreasing cup or glass measuring cup. Pour or spoon off excess fat and discard. Add the chicken broth to the roasting pan and place directly on a burner over medium heat. Add the pan drippings and cook, stirring with a wooden spoon, and scraping up any browned bits, until the mixture is hot, about 2 minutes. Strain and pour into a gravy boat and season with salt and pepper. Carve the chicken and serve with the sauce.