Probably not.
Claiming land by adverse possession or "squatter's rights" as it's also known requires a pretty exact forumla you have to follow.
One of the key elements in that formula is that the possession has to be what's known as: "open, notorious, and hostile" which means you have to occupy the property and prevent the current legal owners from using or coming onto their property.
If you're maintaining it with the owner's consent you would fail that test.
There are several other elements as well.
I believe you would have a pretty tough time claiming some city property you're just maintaining (I assume mowing, trimming brush, picking up trash, etc.) especially since they've given you permission and they know you're doing it.