Pour chicken broth into a large stockpot over medium-high heat; add squash, celery, apples, leek, garlic, allspice, and thyme. Bring the mixture to a simmer and cook until the vegetables are completely tender, about 30 minutes.
While the vegetables simmer, put parsnips into a large bowl. Pour coconut oil over the parsnips and toss to coat. Spread parsnips onto a baking sheet.
Roast parsnips until tender and browned, about 30 minutes; add to soup.
Ladle vegetables and liquid into a blender in batches so the blender bowl is no more than half full. Hold lid in place with a towel and pulse blender a few times before leaving on to blend; puree until smooth. Repeat until all soup is pureed.
Pour pureed soup into the pot, stir half-and-half into the soup, and cook until hot, about 5 minutes.
