What are the rules and how strictly are they enforced? E.g. flying over a park in Paris is off limits? Or anywhere in a city unless over private property with the owners permission?
It's prohibited to fly over cities, airport, heliports, military areas, nuclear plant, national parks (regional parks are allowed). When flying is allowed, maximum altitude is 150 m and you should have always your drone on sight of view. You can check the different areas (prohibited and allowed with athorized altitude) on this official map: Géoportail or this one Mach7 (Mach 7). You just have to enter the place you want to fly over and you'll get all information about what is allowed or prohibited.
In France, it is allowed to fly over private property without the owner permission. Airspace in France is regulated by DGAC, the equivalent of FAA (in the US)
Have a nice trip in France.