In a medium bowl, whisk together 1 tablespoon olive oil, soy sauce, lemon juice, red wine vinegar, dried oregano, smoked paprika, garlic powder, ground cumin, black pepper, and salt to create the marinade.
Add the sliced tempeh to the marinade, tossing 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, lemon juice, minced garlic, salt, and fresh dill in a small bowl. Mix well and refrigerate until ready to serve.
Heat the remaining 1 tablespoon olive oil in a large skillet over medium heat.
Add the marinated tempeh strips to the skillet and cook for 3-4 minutes on each side, or until golden brown and slightly crispy.
Warm the pita breads in a dry skillet or oven for 1-2 minutes until soft and pliable.
To assemble each gyro, place a handful of shredded lettuce onto the center of a pita bread.
Add cooked tempeh strips on top of the lettuce.
Top with cherry tomatoes, cucumber slices, red onion, and chopped parsley.
Drizzle the vegan tzatziki sauce generously over the fillings.
Fold the pita around the fillings and serve immediately.