In a medium bowl, combine smoked paprika, ground cumin, dried oregano, garlic powder, onion powder, black pepper, and salt.
Add 1 tablespoon olive oil to the spice mixture and stir to form a marinade.
Add the sliced tempeh to the marinade and toss well to coat all pieces evenly. Let marinate for 10 minutes.
While the tempeh marinates, prepare the vegan tzatziki sauce by mixing vegan plain yogurt, grated cucumber, lemon juice, minced garlic, salt, and black pepper in a small bowl. Stir 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 strips to the skillet and cook for 3-4 minutes on each side, 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 the gyros, spread a generous amount of vegan tzatziki sauce on each pita bread.
Top with cooked tempeh strips, diced cucumber, halved cherry tomatoes, sliced red onion, and chopped fresh parsley.
Fold the pita around the fillings and serve immediately.