Thank you for your question. The Extension Agent suggested squirrels might be eating your peaches.  When I look at pictures on the internet (fruit eaten by squirrels), the patter is very similar.  See below - information about ways to protect your trees from squirrel damage.  Also, if you have bird feeders in your yard you may wish to consider removing them temporarily.

 http://texasforestservice.tamu.edu/uploadedFiles/TFS_Main/Urban_and_Community_Forestry/About_Urban_and_Community_Forestry/Urban_Forest_Information_Sheets/Tree%20Health%20-%20Injury_Squirrel%20and%20Other%20Small%20Animals.pdf