Combine the parsley, thyme leaves and olive oil in a small mixing bowl. Whisk briefly until smooth. Brush the tuna steaks with the oil, then cover and chill.
Melt the butter together with the olive oil in a large saucepan set over a medium heat. Sweat the onion for 8-10 minutes, stirring frequently, until it starts to soften. Reduce the heat a little and continue to cook the onions for a further 10-15 minutes until they start to caramelise.
Meanwhile, heat a griddle pan over a high heat until hot, then reduce the heat to moderate. Sear the tuna steaks for 1-2 minutes on both sides, brushing with more of the oil as they cook. Reduce the heat some more and continue to cook for a further 6-8 minutes until they are cooked through. Remove from the heat and allow to rest for a few minutes wrapped in foil.
Spoon the onions onto serving plates and season with the red and black pepper. Cut the tuna steaks into big strips and sit on top of the onions. Garnish with sprigs of thyme before serving.