I just looked it up , and it turns out you can get a permit to search for a specifically lost item in Austin. (like if you lost your own ring). If you google "Austin, TX" + metal detecting, it's currently the 2nd hit on the page. In pdf. form.
Couple of thoughts on this :
a) I have a sneaking suspicion of how this "off-limits" status probably historically arose in Austin, TX. Care to take a guess ?
b) What's to stop you from loosing your ring, and simply applying for the permit to find it ? Heck, I loose a LOT of rings.
And you can have your buddy run a "lost" ad on C.L., print it out, show that you are being a good citizen by helping the poor destitute person. And , he/you aren't *quite* sure where it went missing in the park. So you have to do wide search patterns. But not to worry : You have the "permit" that allows you to search now. Eh ?