I've been to US regional parks a few times and managed to get a grill and picnic table for free, without reservation.
At the same time, when I look up park websites, for example Cabin John Regional Park, picnic shelters cost substantially:
Per 1/2 day morning (9am-2pm), residents $80
So do I have to pay and/or reserve for picnic tables and grills or not? If not, is there a way to look up the availability of these free tables and grills?
I looked over the website you provided. It appears that these permits are for large events. It reads like the park might have large picnic pavilions. I would guess that the permit and fee would be to reserve the entire pavilion for a large private gathering.
Though this may be a simple solution, call the park office to confirm availability and fee info.
The vast majority of picnic sites are available on a first-come first-served basis. This is true across the many local, state, and federal agencies that provide picnic sites. These sites are for individuals or single families. There is usually a single picnic table and sometimes a fire pit or grill per site.
Some picnic areas are part of official parks that require a entrance or day-use fee. There is usually no additional fee for using the picnic area once you are already there. This kind of arrangement is common for state parks where there is some attraction and reason for wanting to be there other than to just stop in and have a picnic. Roadside picnic areas are far more common, and generally free.
The page you link to seems to be for reserving a large picnic area for large groups. Gathering groups takes planning, so officially reserving space makes sense. Such things are likely to require a fee, in part because people have to get involved with the registering and reserving, and in part to limit the number of requests to a reasonable value. Sometimes the same park will have a few free first-come first-serve sites, and larger pavilions or group sites that require advance reservations and a fee.
