The Prince George's County Board of Education approved a controversial plan last night to close 31 elementary schools over the next three years as about 350 parents, seeking unsuccessfully to save their neighborhood schools, looked on.

Combined with a plan approved Feb. 11 to close 13 junior highs by 1985, it is the most extensive school closing plan in the Washington area.

The board also approved final changes in its $309.8 million budget for the next school year, asking County Executive Lawrence J. Hogan for $13 million more than he says the county can afford.

The votes to close schools as an economy measure drew the large crowd of parents, many of whom were carrying placards opposing the closing of their neighborhood schools.

The school board has been challenged to close schools, eliminate programs such as junion high sports and take other economy measures because of the fixed ceiling of TRIM, the county charter amendment approved by voters that limits property tax collections, plus a decline in budgeted federal impact aid from $10 million two years ago to zero this year.

The board carried through on the closing plans drafted by its staff despite 10 motions from four of the nine board members to either exempt some schools from closing or postpone the plan for further study. The closings are expected to save $3 million next year and $35 million by 1985.

Many parents, who turned out in the hundreds at public hearings last week, felt the school officials' criterion of closing older schools with small enrollments and consolidating with nearby schools was unfairly applied.Parents from one school, Crestview Elementary in Clinton, said the 1973 court-ordered busing plan had unfairly stripped their school of students. School officials maintained that it is declining enrollment and not busing for desegregation that is responsible for the closing.

At one point, some parents from Hyattsville complained that the closing of the school to which their children are now bused would not return them to a neighborhood school, since the plan calls for them to be bused to another school. Board member Norman Sanders responded that "because of the flight of all races, this school system is resegregating itself right now."

The mood at last night's board meeting was tense. As each vote was taken, parents from the affected school moved to the edge of their seats, hoping in vain for a reprieve.

The budget, which will be submitted to County Executive Hogan today, has been called a "crisis" and a "bare-bones" spending plan by County School Superintendent Edward J. Feeney.

Hogan has said the school budget should not exceed $297 million, allowing 3 percent growth over the current year. The board's $309.8 million budget includes $14 million reserved for a settlement with the county's 7,000 teachers, whose contract talks with the county are at an impasse.

School officials say they have no idea how their budget will be adjusted by Hogan and the County Council, but they feel certain that cuts by Hogan and the cost of a teacher settlement will force cutbacks in some educational programs.

The board's series of votes on closings scheduled the following elementary schools to close next September: Accokeek, Beaver Heights, Berwyn Heights, Buckingham, Camp Springs, Meadowbrook, Samuel Morse, Surrattsville, Thomas Addison, Tanglewood, Forestville, Greenbelt North End, J. Enos Ray, Parkway, Ritchie Beltsville, Chestnut Hills and Powder Mill.

The following elementary schools are scheduled to close in September 1982, although subject to review by the school board next year: Andrews Air Force Base, Crestview, Riverdale Hills, Rosecroft Park and Tall Oaks.

The following elementary schools are scheduled to close in September 1983, although subject to review by the school board in 1982: Colmar Manor, Kentland, Harmony Hills, Landover Hills, West Lanham Hills, Margaret Edmonston, Woodley Knoll and Sandymount.