No permits needed on public parks, but you have to make sure you know which ones are public. I would try Woodland Park. It was opened around 1900 and used to have a small lake n the middle where people swam. It's been hunted more than once but not so much with the newer, deeper machines. There's another one called Castlewood that has an old mansion from the 1800's on it. Been hunted but.................. There's a small park next to the Transylvania University called Gratz. Due to it's size that one has been covered in years past. No one knows whether it's owned by the city or not and locals use it for dog dumping.
You would probably do better by knocking on doors and searching front yards. Your best bet would be to go to a nearby town, either Nicholasville or Versailles. Each town is within 15 minutes, has a park with old history, been hunted less, and has tons of old houses from the 1800's.
Good luck. Look me up when you get to Lex and if I'm not working I will go out with you. I know a couple of nice spots. Been detecting Lex for many years.