(On a personal note I leave for Belgium and France in a few days, but still undecided if I will visit this park as my only zoo of the trip. If the train stopped within walking distance I would go for sure, but the nearest station - Marles-en-Brie - is six miles away and I don't know if I want the bother of trying to prearrange a taxi.)
There is a bus from the town of Melun, easily reachable from Paris. This is how I visited in 2016.