Peonies are usually propagated from the root, but it is possible to grow them from seed. This is how the plant propagates naturally, and is the method always used by plant breeders to develop new varieties.
The first step is to allow some blossoms to set seed heads. When the seed head is mature, cut it from the plant. Cut to the first cluster of leaves after the blossom. Then cut the seed head from the stem. Use sharp, clean pruning shears for this to avoid injuring the main plant.
Place the seed head in a Ziploc bag with moist vermiculite or other soilless planting medium. Push it to the center of the medium so it is well covered, and close the bag. Store the bag in a warm area of the house, out of direct sunlight, until roots are visible. Keep moist but not too wet.
Once the root has broken through the seed and is about 1/2″ to 3/4″ long, put the bag in a cool place (about 40 degrees Fahrenheit). The refrigerator works well for this. Allow the seedling to remain in the cool area for about 3 months, until stems appear. They will be white like the root just small branches leading off the main root.
After the stems have emerged, plant the seedling in germination mix in a pot. Keep the seedlings under strong grow lights until time to plant outdoors. Peonies are planted in the fall to give them plenty of time to grow a strong root system before the summer growing season. Before transplanting, harden off the seedlings by leaving them outdoors for gradually longer periods of time.
Prepare the soil several days before transplanting by working deeply and mixing in plenty of compost. If the soil is alkaline, mix in rhododendron planting soil, which will supply some acidity. Peonies prefer slightly acid soil (6.5-7). It will take 5-7 years for peonies to mature from a seed. Use patience.