Hi Anne!  You may be able to grow new peony plants from the seed pods provided the peony is not a hybrid. If you know the variety you bought and it has ITOH in the name, it is a hybrid and the seeds will be sterile. Many garden shops sell primarily ITOH peonies but you might have a different variety. Note, the peony you get from the seeds may resemble the parent but might not be an exact duplicate.

I believe a lot of people harvest their peony seeds near the end of August. I would wait until the pods are brown and dried before clipping off. There are a number of sites on line when you google Peony Seed Pods which can give you information on harvesting, processing and planting peony seeds.  While I can't recommend one site I did find good information at treepeony.com

I hope this helps you with your peonies and good luck with your seeds!

Douglas County Master Gardeners