start by approaching the park officials--superintendent, park board, someone like that. Get as much good info about DG as you can--check out the front page of the PDGA website, there has been an article linked from there which is great!
Keep it casual at first, just a "what if" kinda thing. If they don't run and hide at first query, start getting a bit more serious. That's when a layout can start to take shape; it may change if they don't want particular areas used for whatever reason, or want OTHERS used. Emphasize the FREE nature of playing, the family possibilities, the increased usage of the park (which can be seen as good AND bad, so stress the positives--trash pickup by DGers, safety in numbers, etc) and the emphasis upon nature and the environmental friendliness of the organized game.
It could be a bit of an economic boost too...maybe only a Coke or fast food meal per person, but that's something.
These are some of the things I used in Ste Genevieve MO
Hope they get you in the door at least...holler when/if you want/need more help, including design ideas!