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Pat the cleaned sand dabs very dry with paper towels.
In a shallow dish, whisk together 0.5 cup all-purpose flour, 0.5 teaspoon salt, and 0.25 teaspoon black pepper.
Lightly dredge each sand dab in the seasoned flour, shaking off any excess.
Heat a large heavy-bottomed skillet over medium-high heat and add 2 tablespoons olive oil and 2 tablespoons unsalted butter. Let the butter melt and get foamy but not browned.
Once the fat is shimmering, place a few sand dabs in the skillet without overcrowding and cook for 2 to 3 minutes per side, until golden brown and the flesh is opaque and flakes easily (internal temperature 145°F).
Transfer cooked sand dabs to a plate. Add the remaining 2 tablespoons unsalted butter to the skillet and let it melt before cooking the second batch the same way.
To serve, sprinkle the sand dabs with 2 tablespoons fresh chopped parsley and accompany with lemon wedges. Serve immediately.