When they are first planted outside, parsley plants need regular watering--an inch of water a week from rain or from a sprinkler--until they become established.
If you have good soil with lots of organic matter and if you mulch the plants, water them only when it has not rained for a week or two. If you have poor soil with little organic content or plants are not mulched, then you may have to water the plants every sunny day, at least until they are well established. This is especially true for parsley growing in containers.
A soaker hose system hooked up to an automated watering timer is very effective for delivering water to the herb patch.
Parsley is a heavy feeder and benefits from some supplemental fertilizer even if it is planted in good soil. When preparing to plant seeds or seedlings, incorporate some all-purpose slow-acting granular fertilizer into the soil to provide steady, consistent nutrition over the season.
If the soil is poor, you may want to also spray parsley foliage with diluted liquid fertilizer or plant tonic once a month or so to insure healthy growth.
Parsley can be affected by competing weeds so mulching parsley helps retain water and keeps down the weeds. Spread 2 to 4 inches of some organic material such as chopped leaves, hay, or straw on the soil around parsley plants when they are about 6 inches tall.
Use peat moss as a mulch only in combination with coarser materials such as chopped leaves or shredded paper. Used alone, it tends to crust over and repel water. As the mulch decomposes in the summer heat, add more to maintain optimal mulch depth.