Pat the cleaned sand dabs (or plaice fillets) dry with kitchen paper.
In a shallow dish, whisk together 60g plain flour, ½ tsp salt and ¼ tsp black pepper.
Lightly coat each fillet in the seasoned flour, shaking off any excess.
Heat a large, heavy-based frying pan over medium–high heat. Add 2 tbsp olive oil and 2 tbsp unsalted butter. Let the butter melt until it foams but doesn’t brown.
Once the fat is shimmering, add a few fillets to the pan without overcrowding and cook for 2–3 minutes on each side, until golden and the flesh is opaque and flakes easily (internal temperature 63°C).
Transfer the cooked fillets to a plate. Add the remaining 2 tbsp unsalted butter to the pan and let it melt before cooking the next batch in the same way.
To serve, sprinkle the parsley over the fish and serve with lemon wedges. Serve immediately.