They absolutely did not have the golden ratio in mind.

In principle it's possible that a notion of the golden ratio was known in some form at the time the Parthenon was constructed in the mid-400s BCE: but very unlikely. For one thing, it's entirely unattested in that period. And for another, the first investigations into irrational ratios were made by Theaetetus around the year 400. Actual approximated values of irrational numbers weren't a thing until much later.

But you're right, the overall shape of the temple is 4:9, which is a ratio of 2.25, a long way from 1.618... There are other proportions in the temple, of course: but they're rational ratios like 2:1 (= 2), 7:3 (= 2.333...), and in some parts of the Parthenon, 81:30 (= 2.70). For accurate reports on the ratios in the building, see Lehman and Weinman, The Parthenon and liberal education (2018), pp. 61-104.

The earliest I've been able to trace the myth that there are golden ratios embedded in the Parthenon is Jay Hambidge's 1924 book The Parthenon and other Greek temples. There's a condensed/super-crazy version in the introduction by L. D. Caskey, which comes across as having a flavour very like the ravings of a modern conspiracy theorist. Every claim that he makes there about the proportions of the temple is false.

Here's a modern counterpart of the same kind of thinking. You really have to work hard to find golden ratios in the facade of the Parthenon! And it has to be said: if you're finding it in ratios like φ^(3)√5 and 10π/3, you're not actually finding it, you're making it up.

As well as that, the measurements are mostly wrong, which is often a problem with efforts along these lines. For accurate figures on the Parthenon, you need to refer to Orlandos' three-volume work Η αρχιτεκτονική του Παρθενώνος (1976-1978); a selection of measurements are given in English by Lehman and Weinman, pp. 167-168.

There is a sister myth, that the golden ratio is apparent in the work of the sculptor Pheidias, who was the manager overseeing the building of the Parthenon (but not its architect: that was Iktinos/Ictinus). It was supposedly in honour of Pheidias that Mark Barr assigned the letter φ (the first letter of Pheidias' name) to the golden ratio in the 1910s (Theodore Cook, The curves of life, 1914, p. 420); though later, in 1929, he's on record as saying he didn't believe Pheidias actually used the golden ratio.

The symbol φ given to this proportion was chosen partly because it has a familiar sound to those who wrestle constantly with π (the ratio of the circumference of a circle to its diameter), and partly because it is the first letter of the name of Pheidias, in whose sculpture this proportion is seen to prevail when the distances between salient points are measured. So much is this the case that the φ proportion may be fitly called the 'Ratio of Pheidias.'

The modern popularity of the myth is, I believe, closely tied to the Disney cartoon Donald in Mathmagic Land (1959), which made a point of imagining golden rectangles in various parts of the facade.

To the Greeks the golden rectangle represented a mathematical law of beauty. We find it in their classical architecture. The Parthenon, perhaps one of the most famous of early Greek buildings, contains many golden rectangles. These same golden proportions are also found in their sculpture.

Here's a few snapshots. In the real building, none of those are golden rectangles. In the first image, for example, the peak of the Parthenon's pediment is much too low to match the profile of a golden rectangle; there and in the second image, the rectangle around each pair of columns and the space in between is 4% taller than in the real building. (It's easier to superimpose golden rectangles over an image if the image is hand-drawn, I guess.)

Caskey does similiar things, claiming that the top part of the building, from the entablature upwards, has the proportions of two golden rectangles side by side; in reality it's much wider than that. I've seen claims that each metope plus triglyph together have the proportions of a golden rectangle; in reality it's a 5x3 rectangle, where the triglyph is 2/3 the width of the square metope. Each metope-plus-triglyph is within 0.25% of the 5x3 rectangle, which is pretty good precision, so it's not as though they couldn't have made a golden rectangle if they'd actually wanted to. (If a golden rectangle had been the aim, they'd be 2.7% off.)

This is a short-form summary of a piece I wrote offsite in 2019.

For the record, there are genuine cases in history of people modelling things after golden rectangles. The most notable example is Salvador Dali's painting The sacrament of the last supper (1955), which takes inspiration from the book that popularised the golden ratio in the modern period, Pacioli's Divina proportione (1509). The canvas is within 1% of being a golden rectangle, the dodecahedral framework in the background of the painting is modelled on one of Leonardo's illustrations for Pacioli's book (and pentagons and dodecahedrons actually do contain golden ratios!), and the figures in the painting are in groups of Fibonacci numbers (which have the property that their ratio approaches the golden ratio as the sequence goes on).

Other than really in-your-face examples like that, it's best to be extremely sceptical when fans of the golden ratio try to convince you that it's everywhere.