Mix lettuce, tomatoes, cucumber, purslane, mint, and parsley together in a large bowl. Place in the refrigerator.
Whisk 1/4 cup olive oil, lemon juice, red onion, pomegranate molasses, sumac, and salt together in a small bowl to make dressing. Place in the refrigerator.
Heat 3 tablespoons olive oil in a large skillet over medium heat. Cook and stir pita squares in the hot oil until golden brown and crunchy, about 5 minutes. Remove from heat and let cool.
Pour dressing over salad; toss to mix. Scatter pita squares on top.
