I voted for the Beach Boys, but I wouldn't necessarily classify them as "rock." But when you're talking about full song harmony, you're looking at either the Beach Boys or CSNY - sometimes the Eagles (Seven Bridges Road).
If you dig any of the three bands I mentioned and also dig full song harmony, you should check out
The Thorns. Matthew Sweet, Shawn Mullins and Pete Droge in a full, all-harmony-all-the-time band. They are hopelessly derivative of your classic 70s bands, but there's no denying how catchy and sweet their harmonies are.