I think that a lot of the claims are quite believable.
The measurements of the Parthenon are quite objectively due to phi. As you can notice in the image below, all major measurements are strictly related to the golden ratio.
I find way more believable that the Greek would have used some symmetrically pleasing geometric construction, such as phi, to design the Parthenon, than assuming it's a mere coincidence.
Regarding the Renaissance, the four artists that extensively used the golden section during the Reinassance are:
Leon Battista Alberti (1404-1472)
See this design and the following quote:
Di questi numeri si servono gli architetti non confusamente e alla mescolata, ma in modo che corrispondano e consentano da ogni banda all’armonia
What he means is that architects should use integers to scale the various parts. To be honest he never explained how he chose the small integers used in his designs so that the composition was harmonious, but subsequent studies have shown he must have used the golden ration extensively.
Piero della Francesca (1416-1492)
A very notable painter, he was also a famous geometer at the time and wrote a book on perspective and proportions which was heavily referenced by Pacioli in his magnum opus. I cannot find an online version but it's cited on Wikipedia (FWIW).
Luca Pacioli (1445-1517) He literally wrote the book on the golden ratio.