Start with a large pot that has a tight-fitting lid. Add the fresh herbs, the pieces of chicken (still on the bone), the garlic, the roughly chopped carrot and celery, and the quartered onion. Over this, pour the cold water. Put the lid on the pot and turn on the heat.
Bring the pot to a boil and then lower the temperature to a simmer. Depending on your stove top this will take around 30 minutes over high heat. Keep the pot covered and cook for 1/2 hour.
Keeping the simmering liquid going, remove the chicken from the pot and set aside.
Remove the meat from the bone and take off the skin. Put the skin and bones back into the simmering pot. Chop up the chicken into very small pieces and put in the refrigerator for a later step.
Re-cover the pot and keep simmering the chicken bones and other ingredients for 2-1/2 hours more.
At this point, remove the solids, leaving only the chicken broth. DO NOT SEASON YET! Future steps will reduce the liquid; seasoning now would end up making the soup too salty!
Add the finely diced carrot, finely diced celery, and wild rice to the simmering chicken broth. Leaving the pot uncovered, simmer for at least another hour until the liquid is reduced by half. This can take a while depending on your stovetop; the last time I made this dish, the reduction took about an extra hour and fifteen minutes.
Stir in the minced parsley and the chopped, cooked chicken that you set aside, earlier. Season with the salt and pepper. Simmer for another 10 minutes and remove from the heat.