Hi,
If you have a portable tabletop grill, then don't mount it on a picnic table. If the Park District Police come riding through, they will ask you to move it to the ground. This is especially unattractive when it is hot and the legs a bit wobbly. I am speaking from personal experience.
If the park is divided into picnic groves, then you will need a picnic permit. You can check if your park falls into the permit system by consulting
this page. If a grove/area is reserved and you are not the permit holder, then you will be asked to move on by the Park District Police.
For the cement area between two buildings, you need to talk to your landlord. It is not only his property, it is his neighbors.
Regards,