The first calls of 1977 to National Park Service offices here, beginning promptly at 8 a.m. Monday and continuing almost nonstop ever since, have been to reserve summer picnic area at Fort Hunt, near Mount Vernon, and other local federal parks popular with many District residents.

Within only a few hours after the calls began, every Saturday and Sunday in June, July and August was booked for Fort Hunt's largest picnic ares, with a closed pavillion and space for 500 picnickers.

Carderock, in the C & & Canal National Historical Park, got off to a slightly slower start, with its largest picnic area and pavillion quickly reserved for every weekend in June, every Saturday in July and every Sunday in August.

Fort Washington, on the Potomac River in Prince George's County, does not open its phones for reservations until Feb. 1, but its reservable picnic areas are among the most popular in the Washington area, especially with district residents. More than 1 million picnickers a year descend upon Fort Washington's reserved and unreserved picnic areas, frequently forcing Park Police to close the fort's gates.

While most picnic and camp sites in the 300 national parks are available on a first-come, first-served basis, some popular parks like those around Washington, Acadia National Park in Maine and a number of Western parks like Dinosaur National Monument in Utah do permit reservations. The Park Service began a nationwide computerized reservation system several years ago but dropped it as too cumbersome.

A number of state parks, like Maryland's Assateague Island Park, also permit advance reservations. The Asateague camp grounds, with 311 camp sites, received at least 60 mail reservations on opening day of the 1977 booking season. The federally owned Assateague Island National Seashore Park, just south of Maryland's park, has about 150 camp sites but permits no reservations.

In the Washington area, reservations can now be made for Park Service picnic areas in Rock Creek Park and Fort Dupont in the District, at Carderock and Greenbelt Park in Maryland and at Prince William Forest Park in Virginia. Prince William also permits campground reservations, while camping sites at Greenbelt and Catoctin Mountain Park in Thurmont, Md., are first-come, first-served.

To reserve picnic areas by telephone, area residents may call Fort Hunt at 557-8991, Fort Washington at 763-1770, Carderock at 299-4159, Greenbelt at 344-3948, Rock Creek Park and Fort Dupont at 673-7646, and Prince William Forest at 703-7181.