When I went to Parc Felins the first time, exactly one year ago, I took the RER from Paris Magenta station (which is connected to the Gare du Nord where the Eurostar arrives) to Tournan En Brie. From there I took a taxi to Fontenay-Tresigny, a charming small town that has the closest hotel to the park. I then took a taxi from the hotel to the park and back again, which added up and was a bit of an inconvenience.
I saw this year, however, there is a bus stop on the road outside the park and I did see a couple visitors use it, so it may be possible to get there by bus.
However, I decided this year it would be easier to rent a car. This was far easier going from hotel to park (and if you want to explore the surrounding countryside - I did a quick drive to the very nice looking town of Collumiers). But the bad part - you Europeans are going to laugh your heads off at my naivete - is that I was dumb enough to rent my car at the Gare du Nord and drive in downtown Paris. Getting out of town was actually not that bad (although I got lost twice on the way to Fontenay because I had not printed clear enough directions from Google). But returning the car was a nightmare. It took me one hour to get from the freeway offramp to the (completely unmarked) rental return lot - which is only a distance of one mile. If I had to do it again, I would take the train to an outlying town that has a rental car near the train station, or try to figure out how to get there on bus.