In a medium bowl, whisk together 1 tablespoon olive oil, lemon juice, dried oregano, smoked paprika, ground cumin, garlic powder, onion powder, black pepper, and salt to create the marinade.
Add the sliced tempeh to the marinade and toss to coat evenly. Let it marinate for at least 15 minutes.
While the tempeh marinates, prepare the vegan tzatziki sauce by combining vegan plain yogurt, grated cucumber, minced garlic, lemon juice, fresh dill, and salt in a small bowl. Mix well and refrigerate until ready to use.
Heat the remaining 1 tablespoon olive oil in a large skillet over medium heat.
Add the marinated tempeh slices to the skillet in a single layer and cook for 3-4 minutes on each side until golden brown and slightly crispy.
Warm the pita breads in a dry skillet or in the oven wrapped in foil for 2-3 minutes.
To assemble the gyros, place a few slices of cooked tempeh on each pita bread.
Top with cherry tomatoes, red onion slices, diced cucumber, and chopped parsley.
Drizzle generously with the vegan tzatziki sauce.
Fold the pita bread around the fillings and serve immediately.