According to County regulation, "An exemption to this requirement (can be) granted by the Administrator upon the written recommendation from the Owner’s veterinarian."
The Administrator of this regulation is Cook County Animal and Rabies Control, so to get an exemption, you must send a letter to CCARC  along with a letter from your vet explaining why your dog is not vaccinated. He will review it and decide if you can still use the dog parks. This may take some time, so if you need an exemption, you may want to get started on it. I would suggest that since you are asking for a favor, you keep the commentary to a minimum.

Requests for vaccination exemptions can be sent to:
Cook County Animal and Rabies Control
10220 S. 76th Ave.
2nd Floor
Bridgeview, IL 60455

My dog doesn't