Uninvited guest is a relative

Just politely write a letter or e-mail and explain that you have a very tight schedule and also busy with your children and work.  Tell them that you don't want to disappoint this 14 year old but he won't enjoy it in your home because he won't get the attention that he should get as a guest.  Not unless the child will help you with the household chores, he's welcome to come.  You'll see if he'll come.  The moment the child learns that he's expected to help around the house, possibly he will tell his parents he does not like the vacation at your house.

Well, it's kind of late to cancel the plane ticket.  I don't know how they got the idea to send their child to you without asking and making plans first.  You have two choices, tell them to cancel the ticket and the trip, or make room for the child to sleep, and cook extra food for her to eat.  The rest of the time she is on her own for amusement and supervision.  She is 14, which is old enough to take care of herself, and possibly babysit and clean house, too.  Who knows, if she is useful enough, you may want to keep her permanently.

Some people you just have to be blunt with, they may be some of these people. So tell them bluntly why he or she isn't welcome at this time., but leave the door open for some other time.