I agree with the above to get a clearance. Park authorities are not paid to interpret but enforce rules and regulations.
Remember that the new permit is most likely based upon two points:
1) Commercial photography, typically, interferes with the full enjoyment of the park by others; and
2) If an enterprise is making money off a city facility, then the city should also share in those revenues.
I've run into this all the time and usually the park employees are given instructions by their supervisors that the size and type of camera is a qualifier in the determination of 'commercial photography'.
When/if you contact the city, make sure you get the name of the person providing you with a detailed definition of the regulation. When/if you get clearance to proceed with no permit, it would be good to back your words, (should you be confronted), with the name of the city employee/manager providing you said clearance. An email would be even better. In your discussions with the city, be prepared to clarify that 'commercial' photography is a legal/accounting/tax definition and has no bearing on camera size.
This shouldn't be a big deal ... until you run into a city/park employee who is just following instructions based upon their supervisor telling them that size and type of camera dicates if it is 'commercial' or not. Reading the city ordinance regarding the permitting process should/could reveal what the city considers as commercial photography.