Dice the potatoes into 1 cm cubes, and dice the onion and bell pepper to a similar size. Mince the garlic.
Heat the olive oil in a 12-inch cast iron skillet over medium heat. Add the potatoes in a single layer, cover, and cook for 8 to 10 minutes, stirring once or twice, until fork-tender and browned on the cut sides.
Add the onion and bell pepper to the skillet. Cook for 3 to 4 minutes, uncovered, until the onion turns translucent.
Push the vegetables to one side and add the ground turkey to the skillet. Break it up with your spoon and cook for 6 to 8 minutes, stirring occasionally, until no pink remains and an instant-read thermometer reads 165 F / 74 C.
Stir the garlic, smoked paprika, cumin, salt, and pepper into the turkey and vegetables. Mix everything together so the spices coat the whole pan.
Use the back of a spoon to make 4 wells in the hash. Crack an egg into each well.
Cover the skillet and cook for 4 to 5 minutes, until the egg whites are fully set and the yolks are still soft in the center. Cook 1 to 2 minutes longer for firmer yolks.
Remove the skillet from the heat, scatter parsley over the top, and serve straight from the pan.