This is a fun ceremony that is a great way to celebrate your marriage. You don't have to be celebrating an anniversary to do this, either.
Call Disney Dining to make a PS at the Garden Grill at Epcot (this is the ONLY Disney restaurant that does this ceremony). I recommend making it at a non-peak meal time; that way your server (who acts as the "minister") is not rushed with serving lots of people. (They will also do this at the Ice Cream Social, I am told.)
When you call Disney Dining, it's almost guaranteed the CM on the phone will not know what you are talking about. That's OK! Politely ask them to note "Rehitching Ceremony" on your PS. (If you want, take a minute to "educate" them.

)
When you check-in at the restaurant, have your DH send mom away (powder her nose or something) to distract her. Then dad checks-in and reminds the CM that they want to do the rehitching ceremony. (Sometimes they might miss the note on your PS.

)
After your meal, your server should announce to everyone within earshot that he/she was not able to attend your wedding so you are going to re-do your vows at Disney to make it official! You'll be asked to stand by your table and repeat your "I do's." Then comes the FUN!!!
You get to kiss while everyone counts to 100!!!!!!!!!!!
They count from 1 to 10 by 1's -- sometimes the server will drag this out a little -- then from 10 to 100 by 10's.
At meals, they may bring you a complimentary cake and you should get a certificate to commemorate the magical moment.
Be sure to bring a camera! Sometimes other servers and maybe even characters participate. It's kinda silly, sort of loud, and just a teensey-bit embarassing ... but it is a terrific magical moment you will remember forever.