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Iranians "claim" to be Aryans because that was the shared ethnic self-identification (Old Persian ariya
, Sanskrit Δrya
, Zend airya
) of the Proto-Indo-Iranian peoples, originally the term for their tribal aristocracy. The modern Iranian peoples are some of their descendents.
However, that is quite different form the "Aryans" that are normally associated with the word. That is a fictitious race invented by white supremacists featuring, in your own words from the other question, "blonde with blue eyes". As such there is no scientific evidence to support the idea that Iranians are the mythical master race of Nazi imagination.
The Iranic peoples are not some sort of "pure" descendents of the Proto-Indo-Europeans either. By the time the Proto-Iranian tribes had migrated to Persia, there were already people there, such as the Mannaeans and the Elamites. Iranians are therefore a mix of different lineages just like everyone else.