Hi Miso,
I don't have a lot of information on Park and Ride within Paris suburbs, but here is what I do know.
Using the Transilien.com site you can discover station information. Take a look at the bottom of this
Transilien page where there is a search box labeled "Les Gares Transilien". Enter "Avon" in the search field and click
RECHERCHER. The resulting page will tell you information on this particular station. Of interest to you would be the column entitled "Stationnement" (Parking in French).
I happen to know this station in particular since I live close by. It has an above ground parking garage, priced at roughly 9€/day (to the best of my memory) and allows multi-day car parking. There is a parking attendant on duty during operating hours (till about midnight), but I'm not sure if there's security after that.
There are additional uncovered parking spots, payable by ticket vending machines, but those are valid up to a 24 hour period maximum.
This station is 40 minutes by train outside of Paris central. You can search for train schedules from Fontainebleau-Avon to Paris on
Transilien.com. I have an article on taking a
train from Paris to Fontainebleau-Avon, but not in the opposite direction, although you may find it useful either way.