Stir together the feta, lemon zest, lemon juice, thyme, and olive oil in a small bowl to combine. Season with salt and pepper, to taste.
Heat 2 tablespoons of vegetable oil in a large skillet over medium-high heat. Toss together the potatoes and parsnips in a bowl. Working in batches, drop spoonfuls of the parsnip mixture into the pan, pressing gently with the back of a spatula to flatten. Cook for 2 to 3 minutes per side, until golden brown. Drain the latkes on paper towels. Continue cooking the latkes, adding more vegetable oil as needed.
Top the latkes with the marinated feta and garnish with micro arugula.