Dormant oil sprays should be applied as close to leaf budbreak as possible. The exact timing can vary from year to year. Early-to-mid-March is probably a good time to start keeping a close eye on those buds. 

Dormant oils can be helpful in managing aphids, peach tree borer (in combo with another insecticide), and can be used to manage pests in other fruit trees.

Here are some resources for you:

Insect Control: Horticultural Oils (Fact Sheet 5.569)

Backyard Orchard: Stone Fruits (Fact Sheet 2.804)

Backyard Orchard: Apples and Pears (Fact Sheet 2.800)

Codling Moth: Control in Home Plantings (Fact Sheet 5.613)