Combine oil, vinegar, pimiento, parsley, green onion, garlic, sugar, basil, salt, and pepper in a jar with a tight-fitting lid. Seal jar and shake vigorously to blend.

Cut cheddar block in half lengthwise, then cut crosswise into ¼-inch slices to form squares. Repeat with cream cheese.

Alternate cheese squares in a shallow baking dish with slices standing on edge. You can arrange them in a single row or in two parallel rows, depending on the size of your dish.

Pour marinade from the jar over cheese, cover, and refrigerate for at least 6 hours or overnight.

Transfer slices to a serving platter so they are still arranged in an alternating pattern. Spoon some of the extra marinade over top, and serve.