Whisk together the lemon juice, vegetable oil, plain yogurt, garlic, tomato paste, salt, oregano, paprika, pepper and allspice in a large bowl; add the chicken and toss to coat. Transfer the chicken mixture into a large plastic bag; refrigerate at least 4 hours.
Preheat an indoor or outdoor grill for medium-high heat and lightly oil grate. Thread the chicken onto metal or wooden skewers. Cook on preheated grill until the chicken is golden and no longer pink in the center, about 5 minutes each side. Sprinkle the parsley over the skewers.
To assemble, divide the quinoa into 4 bowls and top with equal parts of avocado, hummus, garlic sauce, sliced cucumbers and shredded lettuce. Season with salt, pepper and lemon juice if needed. Serve immediately.