If the client has automatic SMS or Email appointment reminders enabled the client's contact will now receive a duplicate of the appointment reminder.
1. What if the client is a child, and doesn't have a mobile, can the reminder just be sent to their parents?
Yes, definitely. In the "Mobile" field for the client (child) just leave this blank. Leave the Appointment Reminder settings for the client on though. Then set their contacts (parents) to receive reminders as described above.