What kind of horse should a parent buy for a child?

You need to consider a few things. Do you think your child is going to ride the horse a few times and then let something else take their interest? If this is the case, then it is probably best not to buy a horse, but instead rent a horse at a riding stable.
If your child has had the chance to take some riding lessons and shows a keen interest in continuing the hobby and talks about wanting to participate in shows and or riding events then it might be time to consider buying a horse.
When you buy a horse the responsibility for it's health and welfare must be shared by both you and your child. Responsibility can be passed entirely onto the child as he or she becomes more mature and understands that horses need daily work and that routines must be established.
If the decision has been made to buy a horse you will want to buy a friendly and well trained gelding or mare. Do not buy a horse from a trail riding stable. These horses are trained to follow a lead horse and can become quite belligerent when taken out on their own.
Buy your first horse from a stable that breeds and trains horses. They may be more expensive, but they won't come with problems sometimes associated with trail riding farms.
Generally speaking horses used for western pleasure riding can include Quarter horses, Paints, Pintos, Palomino's and Appaloosa's to name a few.
Horses used for English riding can include Hunters, Arabians and sometimes Thoroughbreds.
The fact is you should buy a horse that has good personality, mannerism and higher level of training. A three or four year old horse will have enough experience for even beginner riders.

You simply need to choose a breed of horse that you and your child especially like. Once you have determined the breed, start looking for a horse that is friendly, well trained and well kept. A friendly horse is a sure sign of a happy horse that has been trained well and will be suitable for almost any aged rider.