How much you should water peonies depends upon the temperature and type of soil in which they live. New bare root and transplanted peonies need plenty of water during their first year as they become established in their new location.
Peonies are sturdy, but their roots don't grow deep-they grow near the surface, in clumps that can create new plants. As a result, their roots don't pull water from deep in the soil, so they can only reach the moisture in the first few inches of ground. That means watering regularly during dry spells is a must. Peonies are drought-tolerant for short periods, but do typically require an inch of water per week. In the spring and wet seasons, you won't need to water supplementally, but the regular use of a soaker hose during dry times is helpful.
As much as peonies appreciate regular watering, they don't care for compacted, boggy soil,and constant waterlogging can lead to root rot. Watering is a balance, so water deeply once per week.