I'd say it would depend on the park and then on the individual "enforcer". I believe some parks contstrue their permit requirement to indicate that use of gear like a tripod and light stands and such would require a permit, since they are gear that a pro would use and take up the same "space", but as to whether you would be bothered, well, you could check with the park online and see if they have anything posted, you could find someone there to ask, or you could do like the other poster said and just go and shoot, and hope that if anyone does stop you you will have gotten the shots that you needed!
I've seen some pretty elaborate "pro" setups at public parks and I'm sure they had a permit. For the "guy with a tripod", I'd think not. But you are talking about lighting stands and such, more gear involved, so it might be beneficial to find out in advance.