Unfortunately we do not have a large body of pre-17th century Persian Embroidery pieces. Most pieces that do exist even in museums date to the 17th century. According to Phyllis Ackerman in “Islamic Textiles, A History” (part of A Survey of Persian Art), the only examples are a few tirāz bands where embroidery was used instead of weaving.
Persian Islamic textiles seem to have been divided in to two main sections, pre-Mogul Seljuk textiles (ending about mid-13th century) and the Safavid textiles (dating from mid-16th century until mid-17th). Now this includes all textiles not just embroideries, brocades, twills and plain weaves, etc. We know they had textiles and embroideries at this time period, there are records of exports and miniatures showing the use of the textiles. Marco Polo is quoted as raving about the quality of the embroideries of this time. But we can only guess what happen to all of them.
We do have a few pieces that have shown up in the last few years. Two of the pieces are in this handout, a Cloud Collar and an Embroidered Kamis. I have included several more scans of embroidery, one from the 16th and two others from the 17th. These are most likely done in Pattern Darning, I know that the 16th century piece is.