My understanding of the situation is that there are conflicting elements of state law. It would seem that an LTCF would supercede the prohibition of arms in state parks, but this has not been tested and sorted out in court.
Accordingly, the current concensus seems to be NO. The commission of state parks takes to the position that if you are found with firearms in a state park, you'll be arrested unless you have an LTCF, in which case you will be asked to LEAVE. If you do NOT leave, you will be arrested for trespass.