The first calls of 1977 to National Park Service offices here, beginning promptly at 8 a.m. Monday, Jan. 3, and continuing almost nonstop ever since, have been to reserve summer picnic areas at Fort Hunt, near Mount Vernon, and in other federal parks here.

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 area, with a closed pavillion and space for 500 picnickers.

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

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 Assateague 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.

Picnic sites at Fort Washington, the Potomac River fort in Prince George's County, become available Feb. 1. The fort's grounds annually attract about 1 million picnickers.

To reserve picnic areas by telephone, area residents may call Fort Hunt at 557-8991, Fort Washington at 763-1770, Cardercok at 299-4159, Greenbelt at 344-3948, Rock Creek Park and Fort Dupont at 673-7646, and Prince William Forest at 703-221-7181.