I think what I'd do is talk to the Park Maintenance Supervisor, take him out to a grassy, out-of-the-way spot and show how how you dig and replace plugs. Tell him there are different methods that you can use, but start with the big plug -- and then replace it properly. Show him that even a four-inch diameter plug can be replaced with nary a trace, and that this type of plug is actually better than a smaller one, in terms of the root system of the grass. I think if he sees you cut a 4" plug, put the dirt on a cloth, and then pour it back into the hole and then replace the plug, he'd be pretty impressed, and less apt to give you a hard time. You can tell him that if he sees any holes that were not properly replaced, that you can promise him it was NOT you, as you always dig and replace YOUR plugs just as you told him. I'd give him your cell phone number and tell him if he has any concerns, with any "damage" he sees, to call you. IF someone else digs some nasty holes, and he calls you thinking it was you, I'd meet him there, make sure those holes AREN'T yours, and then take him to a few of the (barely discernable) holes that WERE yours, and show him that no, you truly are digging and replacing plugs just as you demonstrated to him -- and show him the proof.
I think after giving him a demo, and then giving him your phone number, you might be in good shape. That would seem pretty responsible and up-front to me, if I were a park manager. If he still doesn't like the plug method, after you demonstrate, then show him the "flap method;" if he still doesn't like it, show him the "slit" method...I think if you demonstrate your willingness to be "flexible" and "cooperative," you will find a compromise; after all, the city did not say "though shalt not," but only "we prefer not," and thus I think there is some room there to work with the park manager and come up with something agreeable to all...
Steve