It says how to make them germinate but it doesn't say how to store them while you're waiting for fall and winter.
How Do I Store My Peach Pits Until Early Winter
Certified GKH Gardening Expert
You can store the dry seed in a plastic bag in your refrigerator provided you do not put apples or bananas in the same fridge (because they release a ripening gas). Alternatively, you can store the pits at room temperature, but out of the sun.