I stayed in NOLA for about a week and had no problem at all just parking in big box store parking lots over nights. I used Office Depot, Lowes, Home Depot and a a few other random ones. Mostly places with wifi.
Speaking of wifi.. The big city park has pretty good wifi. You just have to move around a bit to find the best spots.
Also if you head west on 61, there is a park at a levee, It's called Bonnet Carre Spillway, it's a old US Amry Core of engineers park that is leased to the city. It was originally a campground and there are still signs there talking about camping but the park is free to access. I would assume you could get away camping there for free, but I didn't try as I had no problems just stealthing in parking lots with free wifi.