Combine oil, vinegar, pimento, parsley, green onions, garlic, sugar, basil, salt, and pepper in a jar with a tight-fitting lid.
Seal the 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.
Arrange cheese squares alternately in a shallow baking dish with slices standing on edge.
Pour marinade from the jar over top, cover, and refrigerate for at least 6 hours, or overnight.
To serve, transfer slices to a serving platter so they are still arranged in an alternating pattern.
Spoon some of the extra marinade over top.
