My parents are the same way. I'm 28 and my mom still worries when I drive on the interstate. Here's something to consider.
You may not ever be able to "convince" parents, but you can make them comfortable with your trip. They aren't thinking in logical ways when their child is going off alone. They are imagining violent death, and larger than life accidents. Logical reasons may not work for your particular situation. A comprehensive overview of the safety statistics of a particular hotel or area will not alleviate your parents fear that somehow a rouge mental patient will escape from the asylum and come after you, or that you'll be left out in the cold night when you lose your hotel room key.
Here's a little trick I've used:
Give them a list of emergency supplies you intend to bring. It helps them to know you're responsible and prepared. Include a list of contact numbers of your family and friends. A list of local police and hospital addresses and numbers(you know, in case WDW emergency staff can't get to you, and you have to drive yourself to the hospital
). Make sure your parents know you are bringing your health insurance information and all identification materials you may deem important. Let them know you are also bringing an emergency credit card(or other emergency fund), just in case. Throw in a small first aid kit just to drive the safety point across. It may even help to ask if your parents can help you compile a list of helpful emergency "stuff" you may need. They will feel a bit more sure of your decision if you bring along some things they feel are important to carry around.
One more thing. Beware of the "I'll call you" syndrome. You may get stuck in crowds, or a change of plans that warrant a late or missed call. That may be enough for your parents to send out the National Guard in search. Cell phones may be convenient if you're stuck, but it also gives free reign for unsure parents to call you every second of the day. If you are on a ride, or in a show where cell phones may not be appreciated, and you miss the call, again the National Guard is sent out.
Good luck!
Lurieh