As a family business, may the parent allow their son or daughter to use the license to contract and perform work?

No. Only the approved entity may utilize the contractor license. Family members may be employees of the family business, but they may not use the license belonging to another business or individual.  The license may only be utilized by the owner(s).  It is considered a law violation to allow use of your license to another for contracting, performing work or obtaining a permit.