Pushups are mentioned by the Centers for Disease Control and Prevention as one way to help your child meet her 60-minute-per-day exercise requirements for optimal health. Strength-training exercises, such as pushups, improve a child's fitness level, as well as promote healthy growth of the muscles, bones and connective tissues. Pushups do not stunt a child's growth.
You wouldn't stop a child from playing on a playground -- lifting, swinging and hoisting her own body weight. A pushup is simply a more formal example of this activity. Pushups use your own body weight to strengthen the muscles of your chest, shoulders, arms and core. Squats, pullups, situps and lunges are examples of other body-weight exercises that are not only acceptable but also may benefit children and teens.