Mavericks system tool gets wrong notifications
If you try to log in and the browser fills in the user and passowrd then that is your problem #1

1- Go to the key chain and delete all "your" Facebook accounts there on your dads computer. Be careful not to delete your dads accounts, that might result in a "you are grounded" for a Month penalty.

2- Disable the Keep me logged in in Facebook web page - Browser.



3- In case you have the Facebook set up in sys pref Internet Accounts - delete it.