Can separated parents book their own appointments?
Yes. Each parent or caregiver signs in to your event website with their own email address and registers for the event separately. Their registrations, bookings, reminders and emails are completely separate, and neither can see the other's bookings.
If the event collects students, each parent adds the child to their own profile, so the child appears under both registrations. If the event is invite only, each parent needs their own invite code, because a code can only be used once. The booking limit applies to each parent separately, so both can book the same teacher.
If both parents should attend one meeting together, one parent books and shares the time with the other. Bookings belong to whoever made them, so only that parent receives the reminder and can cancel it.