I don't have official conformation of this, as I have not contacted Starbucks directly, but after getting curious about the new cookie straws they just released that they will be selling in stores to go with the new s'mores frappachino, and eventually other drinks I did some research. I found an article that states that they do potentially contain peanut oil. http://starbucksmelody.com/2015/04/26/the-cookie-straw/
Unfortunately Starbucks does not have information for these listed on there website (that I could find).
So just a heads up to avoid those, just in case.