The Myth:
The Greeks used the golden ratio in the design and construction of the Parthenon. Usually this claim is supported by a photograph of the Parthenon with superimposed golden rectangle (as below, taken from this website). For example, the cited web site states that "Once the ruined pediment is restored, the ancient temple fits almost precisely into a golden rectangle."
The Reality:
Markowsky [Mar92] is probably the best reference I've seen on this one. In short, the chances of this being correct are slim to none. What does appear to be the case is the the ratio 4:9 was used in laying out the height to width and the length to width of the structure (nowhere near 
). Another problem is that the choice about where to draw the rectangle is fairly arbitrary. Why only include part of the platform on which the Parthenon sits, why just some of the steps instead of none or all? If golden rectangles were so important, why not instead choose the rectangle shown below, which at least is part of the structure? From my standpoint the biggest problem with this is that such images ignore the role perspective plays. A rectangle when viewed from the correct viewpoint can be enclosed by a golden rectangle (or any other rectangle you desire). If the viewpoint isn't too far out of true, most people won't even notice the deviation from being a right rectangle. Careful inspection of the figure would indicate that the Parthenon is not being viewed head on in this photograph, and is being viewed from below. A view through the center of the temple parallel to the floor will change the ratio of width to height. Finally, if the architect responsible for the restoration of the Parthenon is to be believed (I wish I could site this, but it was something I saw in an interview on the Discovery Channel), there isn't a straight line in the place. In fact, the pillars were designed with the foreshortening effect in mind, and so aren't cylinders.
From a historical standpoint, it isn't very likely that the golden ratio was even known to the architects responsible for construction of this temple, it is even less likely that they would have attached any unusual importance to it. While it may appeal to some to believe that the Parthenon is beautiful because of some secret geometry, its renowned beauty is more likely due to its effective use of repetition of form and the striking visual quality of the materials used.
References:
[Mar92] George Markowsky, Misconceptions about the golden ratio, The College Mathematics Journal 23 (1992), no. 1, 2–19.
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