Children are in the process of becoming an adult, at which point they are in charge of themselves.
Up to age 6, they have an inner purity, which means that they do not need to practice Tapas. Likewise, they do not study the Scriptures [Svadhyaya], nor perform Ishvara Pranidhana. Karma yoga, and asanas, are not for them. Similarly they have minds that are spontaneous, uncontrolled. Neither the first nor the second chapter of the Yoga Sutras are useful for them.
Up to this age, parents need to maintain the child's purity and not clutter the mind with controlled learning, as otherwise the child's spontaneity is destroyed. The parents, not the child, is responsible for its actions, and therefore they should not be pressurized, nor disciplined. However, nowadays, children are naughty and remedial action needs to be taken straightaway to give the child what it is missing: love.
From the age of 6, the child can practice Tapas, Svadhyaya and Ishvara Pranidhana. The most important things that it needs to learn is ethics [yama] and Tapas. Cleanliness [saucha] and contentment [santosha] also need to be taught along with knowledge of the Mahabharata and the Ramayana.
The practice of Ishvara Pranidhana is something that each individual needs to decide for him or herself: the parents should teach the child all the nine first components of yoga [yama and niyama] and let it decide about its relationship to God. The child should not be brainwashed into a particular relationship.
Hatha yoga asanas will damage a child's development of its sexual energy sphere and should not be taught. When the child is an adult, that is it no longer obeys its parents and takes its own decisions, it is then free to commence asana practice. An exception is when the child has some specific physical condition requiring treatment. Hatha yoga exercises may be required as a therapy. Tantric yoga is good for a child under the care of a father practicing Ishvara Pranidhana. Between ages 6 and 12, the child's sexual energy sphere is under development and Tantric yoga will help to ensure health at a time when disease in this energy is prevalent across the world.