Combine yogurt, mayonnaise, pesto, capers, lemon juice, olive oil, garlic, basil, and parsley in a food processor or blender. Blend on high speed until smooth, scraping down the sides as necessary. Season sauce with salt and pepper to taste.
Place chicken thighs in a sealable bag or bowl and pour half of green goddess sauce over top. Set the other half of sauce aside in refrigerator. Evenly coat thighs in sauce until coated. Seal bag or cover bowl with plastic wrap and allow chicken thighs to marinate in refrigerator 2 hours or up to 12.
Preheat an outdoor grill to 400F OR prepare a large skillet to use indoors on stovetop by greasing well. Remove chicken from marinade and pat dry.
Arrange chicken thighs on grill or in skillet on stovetop and allow to cook, turning occasionally until chicken registers 165F.
Transfer chicken to a plate and drizzle with reserved green goddess sauce. Top with additional fresh basil and serve warm. Enjoy!