Heat 2 tablespoons of the olive oil in a large skillet over medium-high heat. Add the potatoes and allow them to brown for several minutes without disturbing them. Add ¼ teaspoon of the salt, dill, and parsley to the skillet. Stir the potatoes and continue cooking for 3 to 4 more minutes, until crisp-tender.
While the potatoes are cooking, whisk together the remaining 1 tablespoon of olive oil, lemon zest, lemon juice, and remaining ¼ teaspoon of salt. Set aside.
Add the salmon to the skillet and cook until opaque, about 2 minutes.
Remove the hash from the heat and toss with the lemon oil. Serve warm.