You can set individual reminder preferences for specific patients in their patient profiles, including making sure that they only receive reminders by SMS or email or disabling them entirely.
To set individual reminder preferences for a patient:
Navigate to the patient's profile.
In the Reminders section, click Edit this patients' reminders settings.
The Patient Reminder Settings pop-up appears. Untick the checkboxes where you do not want the patient to receive a reminder.
Tick the checkbox at the bottom of the form to ensure that the changes are applied to all future-dated appointments (even appointments already in the calendar) rather than just appointments that have not yet been created in the calendar.
You also need to make sure that your default settings are not overriding their preferences:
Click Settings > Reminders.
Find the level (group or individual practitioner) you want to check and click Edit to the right of the Preferences header.
Ensure that "Patients without reminder preferences" is enabled. This will ensure that your default settings only apply to patients who do not have individual reminder preferences, meaning patients who do have individual reminder preferences will have those preferences apply.