For starters: New Orleans itself has sections of parade routes that are all about families. And Trinidad and Tobago -- which has a stupendous grown-up Mardi Gras -- has a Kiddies Carnival before the main bash.
Families can also catch some Mardi Gras vibes over many weekends at the Universal Orlando theme park: every year, Mardi Gras is celebrated with a 10-block parade, live music, and more on Saturday nights. 2009 dates are Feb. 7 to April 18, and live music includes The Village People, Pat Benatar, Barenaked Ladies, Kelly Clarkson - all part of your regular admission.
For more about Carnival and Mardi Gras Festivals world-wide, see a round-up at About.com's Guide for Italy Travel. Venice is famous for its "Carnivale", but you might not know that Greece and Germany have Carnival celebrations too! And Quebec City has a wonderful family-friendly Carnival.
*Photo courtesy of Universal Orlando.