as you-can-call-me-sir says, it depends on what kind (entity ownership/governance) of park you're referring to. If you have any doubt, look potential rules up FOR YOURSELF (as opposed to asking a desk-bound bureaucrat "can I?"). The reason for this is, you don't want to risk a "no", (simply because some bored desk clerk morphs something else to apply to your "pressing question"), when in fact, no real rule specifically says "no metal detecting". If the rules of the city, county, or state are silent on the issue (ie.: nothing specifically addressing "metal detectors") well then presto, I guess it must not be prohibited
You can check this site, as per the state level parks:
Federation of Metal Detector & Archaeological Clubs Inc.
But even this is highly dubious, as .... quite frankly ........ some of those states with dire sounding "no's" or "inquire at each kiosk" type answers, are ..... truth be told, detected with no problems. The way such verbage gets into a compendium list like this, is that decades ago, someone takes it upon themselves to write a letter to each state capitol's headquarters, asking something like "can we metal detect in state parks there?". Well, duh, what do you *think* the "safe" answer is going to be? Since some parks might be historically themed and sensitive, yet others are probably innocuous and no one's ever cared (regular beaches, etc...), yet you know full well that the "safe" answer is going to be an across the board "no", or whatever. Yet oddly, when lists like this first started appearing, old-timers scratched their heads and thought "since when?". See how that psychology works? It's the old "no one cares, till you ask" routine. For example: If you read of CA on that list, you might think you're burdened with loops to hit state of CA beaches. Yet I can tell you that we hunt state of CA beaches all the time here, in full view of rangers or anyone, and no one ever cares (I guess no one here ever asked enough questions, lest those rank and file would need to look things up in their books, and start booting people).
As for "abandoned" type properties, I hunt them all the time, if I'm travelling around, and see something that looks worth checking. But ..... again, if you want to be technical, it's (as call-me-sir says) technically still "owned" by someone, no matter how "abandoned" it seems. No matter how many short-cut trails criss-cross that vacant lot, or no matter how many fences or signs it "lacks", if you wanted to get "technical", you can't even step off the sidewalk. But again, how technical do you want to be? It may be owned by an out of state bank (bank repo's. or emminent domain, or whatever) and sure...... I suppose you'd have to write letters to some out-of-state bank (and you can guess their response), or inquire of city council to have a meeting about it, pass it through their legal dept, and so forth (and you can guess their response). But ...... to each his own.