For the cupcakes:
For the frosting :
Pro tip: If you don’t have buttermilk or don’t feel like buying it, you can make a substitution by mixing ½ cup milk with 1 ½ teaspoons lemon juice and letting it sit for 5 minutes.
Spray 12 cupcake molds with the vegetable oil, making sure to grease the bottom and sides or drop in cupcake liners. Arrange a rack in the middle of the oven and heat to 350 degrees.
Spread the walnuts out on a baking sheet and toast on the middle rack until lightly toasted, about 8 minutes. Cool. Finely chop half the walnuts and set aside; coarsely chop the remaining walnuts.
For the cupcakes: In a large bowl, whisk the flour, sugar, crystallized ginger, and baking powder. Add the cinnamon, ground ginger, nutmeg, salt, and allspice and combine.
In a separate bowl, whisk the eggs. Add the oil and buttermilk and whisk to combine evenly. Add the egg mixture to the flour mixture along with the parsnips and the coarsely chopped walnuts (reserve the finely chopped walnuts for decoration) and stir until well combined. Divide the batter between the 12 cupcake molds, filling each about two-thirds full. Bake on the middle rack until a toothpick inserted in the center comes out clean, 20 to 25 minutes. Cool the cupcakes on a rack for 5 to 10 minutes, then remove from the tin. (Use a flat kitchen knife to help release them if you haven’t used liners)
Meanwhile, make the frosting: Working with an electric mixer, beat the cream cheese and butter in a large bowl, whisk together until fully blended and light, about 1 minute. Add the vanilla and slowly add the sugar, a few tablespoons at a time, making sure to fully incorporate before adding more. Add the maple syrup and mix until smooth. The frosting will be thinner than a regular frosting; if it feels too thin, refrigerate for about 30 minutes to firm it up, or add more sugar, if desired.
Generously frost the cupcakes on top, sprinkle the top with reserved walnuts and add a sliver or two of the candied ginger.