If using wooden skewers, soak 8 skewers (30cm/12-inch) in water for 20 minutes so they don't scorch on the grill. Skip this if using metal skewers.
In a mixing bowl, whisk together the olive oil, lemon juice, lemon zest, minced garlic, oregano, and black pepper.
Cut the halloumi into 2.5cm (1-inch) cubes. Halve any large mushrooms, leaving small ones whole. Cut the bell pepper and red onion into pieces about the same size as the halloumi.
Add the halloumi, mushrooms, pepper, and onion to the marinade. Toss gently to coat, then let everything sit at room temperature for 15 minutes.
Heat a 30cm/12-inch grill pan, or an outdoor grill, to medium-high heat, about 200C/400F.
Thread the halloumi, mushrooms, pepper, and onion onto the 8 skewers, alternating pieces as you go.
Place 4 skewers at a time on the grill pan so they aren't crowded. Cook for 8 to 10 minutes, turning every 2 to 3 minutes, until the halloumi is golden with char marks and the mushrooms are soft. Repeat with the remaining 4 skewers.
Move the skewers to a plate, sprinkle with chopped parsley, and serve warm.