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 areas at popular parks like Fort Hunt near Mount Vernon and Carderock in the C & O'Canal National Historical Park.

For Hunt's largest picnic area, with a closed pavilion and space for 500 picnickers, was booked every weekend in June, July and August within a few hours after the calls began, while Carderock quickly reserved its largest picnic area and pavillion for every weekend in June, all Saturdays in July and all Sundays in August.

Fort Washington, the most popular Park Service picnic ground near the nation's capital - it attracts about 1 million barbecuers every year - will begin taking reservations for four large picnic sites on Feb. 1.

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 campsites, 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 Caderock 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 703-211-7181.