The Bardenes Reales is a
natural park, which is a different animal altogether from a
National park, also though in this case, like national parks, wild camping is totally forbidden, and there are frequent patrols and very few places to 'hide'. But it's worth being close by if only to get into the area early or late to take better photos.
The only nearby campings are the
Camping Bardenas, which is basically a trailer park. Folks I've known have had crap nights there in the occasional vacant lot but with constant truck traffic more or less all night - truck avoid the tolls on the nearby motorway to drive right past the site.
The other camping that is close is at
Ejea de los Caballeros, which is another holiday resort style site but does at least have dedicated plots, although they are covered in gravel for caravans so sleeping in a tent is uncomfortable, it's also prone to mosquito invasions from the lake that is part of the complex. But the general facilities are good and the people nice.
However the northern approach - to the Shepherd's monument above - passes through mixed woodland where you should be able to find a hidey hole. Although wild camping is technically prohibited anywhere in Aragon and Navarre you can argue that you are making bivouac as long as you pitch tetnt after 20.00 and are off, and I mean off, before 08.00 and that your tent is less that 1.2 metres tall.
There's also a shelter - really a covered BBQ - near the shepherd's monument and I've heard of people sleeping there, but once again, start late and leave early. The parks are patrolled by forest rangers who are usually OK if you show them respect and don't take the piss with wild fires, bottles of whisky, etc.
The points about the trails are well made, but it doesn't rain often ... it only takes a couple of hours* to go right through, if that, so unless you really want to get dawn or dusk photographs then it's probably just as good an idea to just include it in your route- A group of us did just that riding to the HISS a few years ago, starting way up in the Pyrenees at Ochagavia and getting to the HISS - knackered I have to admit! - all way down in Teruel at Terriente and going by back roads all the way ...
Regs
Simon
* it is technically possible to ride at over 100 kph on most of the trails as has been proven in actual tests - but do't ask me how I know this!