The burden of proof for "high impact" is heavy. I recommend reading some of the articles by the Inarah Institute <http://inarah.net/publications> on how shaky the dominant Islamic history narrative is.
To clarify: It is unclear what the direction of the influence is. Was it Islam that impacted/influenced what happened in Persia, or was it the Persian/Syriac/Nestorian culture that created something called "Islam."