Absolutely. The Garden District is a particularly family-friendly place to bring a picnic and watch the parade krewes. You'll want to avoid the French Quarter, which gets even more wild than usual during Mardi Gras. Family Gras in the suburb of Metaire is also a great kid-friendly option. Be sure to bring a bag for all the "throws" your child will collect from the parade crowd. Items might include stuffed animals, toys and beads of course!
The Carnival season begins on Epiphany, which is January 6. Parades take place throughout the days and weeks leading up to Mardi Gras, which is always on a Tuesday. It's a great day to experience the festivities as well as the weekend before, when two exciting parades take place, Bacchus and Endymion. The day before Mardi Gras is Lundi Gras - on Monday of course - and 2 parades, Zulu Lundi Gras and Orpheus, take place.