In a deep sauté, heat olive oil. Add onions and cook over medium-low heat until golden and caramelized, about 15 minutes.
Add parsnips, salt, and water and bring to a boil. Lower heat, cover, and simmer for 20-30 minutes, until the fully tender parsnips have turned a light gold color and most of the water has evaporated. Use an immersion blender to purée until smooth.
Preheat oven to 400°F. Line two baking sheets with parchment paper. Add fish to one baking sheet. Drizzle with olive oil and season with salt and pepper. Bake for 20 minutes.
Add fennel, parsnip, and shallot strips to the second baking sheet. Drizzle with olive oil and season with salt and pepper. Bake for 20 minutes.
To serve, add a generous dollop of parsnip purée to the plate. Spread it out using a spoon. Top with fish and vegetables.