Place the cucumber, red pepper, and lemon zest in a medium mixing bowl.
Pour the dressing over the vegetables. Toss gently to coat. Chill the vegetables and let stand for 1 hour before serving.
For the puree: In a sauce pot, combine the vinegar, sugar, ginger, and lemon juice. Simmer the mixture over medium-low heat until big bubbles start to form and it becomes a light syrup. Reduce until thickened.
Remove the pot from the heat.
Meanwhile, in a large pot, combine the parsnips, potatoes, and chicken stock, and bring to a boil. Cook until both the parsnips and potatoes are fork tender.
Drain the parsnips and potatoes, and mash with the heavy cream, butter, and ginger syrup until smooth. Season the parsnip-potato mash with salt, to taste.
Heat a large saute pan over medium-high heat and add the olive oil, garlic, and shrimp. Cook the shrimp for about 3 minutes, or until cooked through. On a warmed serving platter, make a bed of parsnip puree, top with the shrimp, then top the shrimp with the pickled vegetables and some of the juice.