A Special Flight number confers no status and is more or less a heads up to affected ATC units as well as providing contact numbers for the appropriate agencies to the operator. We normally see them issued for photo surveys and the like.
Parachute paperwork tends to be for individual displays in the form of a Display Authorisation rather than for a season of operation.
I would suggest if it is a regular operation that you get in touch with the controlling authority for the airspace and see if you can agree a permanent Letter of Agreement with them. This will provide both sides with the conditions and procedures to be followed. Quite a few para clubs have these already and maybe the BPA can suggest how you go about it and what kind of things are already in operation with other clubs.