In the bowl of a food processor or a blender, combine pulse parsley, red wine vinegar, garlic, salt, pepper and chile flakes. Pulse until combined, but not smooth.
With the motor running, slowly pour in the olive oil.
The Sandwich:
In a medium bowl, stir together cucumber, hearts of palm and chicken.
Gently stir in the dressing.
Scoop the mixture into the four pita halves. Serve.