To make the salmon: Prepare barbecue for medium-high heat.
Sprinkle salmon with salt and pepper. Rub olive oil, minced garlic and dried oregano over salmon to coat well.
Once grill is smoking hot, brush grill rack with olive oil and place salmon on the grill.
Cook for 4 minutes without lifting or turning salmon. This will help give salmon a good sear of nice grill marks and make it easier to move.
Starting at corner of salmon nearest to you, slide a long metal spatula under salmon, turn it over and cook until salmon is just cooked through and all but the very center is opaque.
Meanwhile, to make the Greek salad: Place the red wine vinegar in large bowl. Slowly add olive oil and whisk continually to blend. Season vinaigrette to taste with salt and pepper.
In large salad bowl, toss lettuce, tomatoes, cucumber, olives and onions with enough vinaigrette to coat. Season salad to taste with salt and pepper.
Divide salad among 4 plates. Crumble feta cheese over salads.
Top salads with grilled salmon fillets. Sprinkle with parsley.
Drizzle some of remaining vinaigrette over salmon, sprinkle with parsley and serve immediately.