You haven't mentioned your age but most of the family annual policies will limit cover for 'children' to those who are under 18 and still in full time education. Some will cover up to the age of 23 but these are rare. Check the defintion of a family before you commit to buy.
If your mum and dad go on a policy together then it is likely that the insurance will cover them to travel seperately from one another but check the policy wording or call the provider to confirm.
Hope this helps.