Brush and wash the parsnips thoroughly, keeping the skins on to retain their flavor. Roughly chop them into chunks. Rinse the chunks in cold water, then place them in a pot. Cover with cold water, add 1/2 teaspoon salt, and bring to a boil. Cook the parsnips until they are fork-tender, approximately 15-20 minutes. Drain the water, reserving 1/2 cup of it to adjust the purée’s consistency if needed.
Using an immersion blender, blend the parsnips until creamy or until you reach your desired consistency. Add some of the cooking water if needed.
While the parsnips are boiling, prepare the onions. In a small sauté pan, heat 2 tablespoons of olive oil over medium heat. Add the chopped onion and the rosemary sprig. Sauté the onions until they begin to caramelize, about 5-7 minutes. Then, add the garlic and cook for one more minute before removing from heat.
Add the onion and garlic mixture to the puréed parsnips, keeping the rosemary sprig for garnish.
Stir in the remaining 1 tablespoon of olive oil, the allspice, and salt and pepper to taste and use an immersion blender to blend until creamy (you can add a bit more of the reserved liquid if necessary).
In the same pan used for the onions, add the chopped hazelnuts with a small drizzle of olive oil and a pinch of salt. Sauté the hazelnuts over medium heat until they are lightly toasted and crispy.
Arrange the parsnip purée on a serving plate. Top with the toasted hazelnuts, the rosemary leaves, a drizzle of olive oil, and additional salt and pepper if desired.