Park rules are available at
http://www.dem.ri.gov/pubs/regs/regs/fishwild/parkmgmt.pdf
I live nearby, and while I haven't gone to Lincoln Woods to fly, it's a pretty crappy rule. I have been working on ways to test the DEM and get it overturned, but surprisingly have not had any park rangers give me any trouble. I have flown at a few other state parks, in plain view of a ranger and they didn't even make a point to get out of the truck to ask questions.
In general, the rule is aimed at curtailing loud noises and nuisances. The phantom would definitely fall under engine powered model aircraft. The rule also mentions being audible at 200 feet. An easy test is to fly straight up until you are 200 feet AGL indicated and see if the noise would rise to a nuisance level. Usually it does not- and I've been in situations where the music coming from people's cars (at Goddard) definitely surpasses that threshold.
The rule also states that you can get a written permission to operate. I believe that riparks.com has a permit application for "special use" and for photography, but no fees listed for either and they don't show up in the list of fees published. A photographer friend confirmed that the permit was free the last time he had to get one to shoot a wedding at a state park.
You can try to get a permit- and if you're denied then you definitely have grounds to fight it since i can't think of a reason they'd deny it other than "just because".
You could also play on the technicality that since they can't regulate flyover and only operation from within the park and try to operate from outside the park. The tunnel under rt.146 near the beach parking lot comes to mind as a spot that would get visibility to the pond.