Many cities, parks and counties across the United States are working on rules, regulations and codes around slackling, which is still considered by most an "emerging sport." This means many law enforcement officers, park rangers and security guards don't actually know what the rules say, or are operating in gray areas when deciding to allow slacklining or not in parks and other public areas. This lack of clarity can sometimes lead to pretty funny interactions with law enforcement.
Is slacklining legal in the USA? In general, in the USA, slacklining is allowed in most cities, parks and public places. Some of the more popular areas for slacklining have some level of regulation, and a few places have all but banned it. The slackline.us website does a great job recapping slackline legality in various "hot spots" (places like like Yosemite and Smith Rock).
Here is another great resource article with links to cities, park departments and counties that have developed slacklining regulations. Check to see if your area has any. And here's one more article (Slacklining is Not a Crime) that does a good job of summarizing the slackline legality issue.
If you're not sure, contact your local city parks department. They can probably advise you on whether any rules apply to the park you're planning to use. There's a good chance that the person you talk to won't even know what slacklining is, or will refer to some obscure code that has to do with "no fixtures attached to trees" (usually pertaining to building tree houses). In such cases you can feel pretty secure that slacklining is OK, as slacklines aren't permanent fixtures.
For most places, this rule of thumb applies: Use common sense. When setting up your line, ask yourself "is this obviously dangerous" or "is this line blocking people from using an area of the park"? If you're setting up a 150 foot line across the middle of a busy city park on a sunny Saturday, you're likely going to run into issues. If you don't use tree protection, you'll run into trouble. If you're doing something clearly dangerous (like a highline between two buildings), you'll run into trouble.
Don't set up a slackline across an area where people are riding bikes.
Otherwise, with a little common sense, it's likely you'll have a problem-free slacklining session (and you'll make it easier for the rest of us when we go to setup our next line).