Find other people in your community who are also interested and would be willing to join a commitee. Then look for a good location for the park. WHen you have a couple ideas for locations begin petitions. Go door to door, set up a stand at the farmers market, go to pet stores, vets etc. Get as many signatures as possible. Then go to parks and rec department. Be prepared to have a good argument about why it is need and a plan for how it will be maintained.
Also think about looking for private funding for things like fencing, trach cans, benches etc.
You will also need volunteers for the council to keep the park up and running.
Hope this helps!