Before you can calculate your payroll tax liabilities, you must first determine which of the people who work for you, if any, are "employees" for whom you must withhold and pay taxes. If none of your workers are classified as employees, you generally won't have to worry about payroll taxes at all!
Each federal and state law imposing a payroll tax has its own definition of the types of workers to which the tax applies. However, as a practical matter, the basic governing standard under all of these laws is whether the individual who performs services for you is properly characterized as an employee, as opposed to an independent contractor, under so-called "common law rules."
In general, these rules say that your workers must be treated as employees if you have the right to direct and control the way they do their work, rather than merely the results of the work.
Once your workers are properly classified as being employees, the fact that they may work for you only on a part-time or temporary basis or that they may be minors generally won't relieve you from the obligation to withhold and pay taxes on their wages. Again, the key issue is whether the workers are common-law employees, and that determination is unaffected by the number of hours the workers put in or by their age.