The Fairfax County Park Authority, which is on the verge of being taken over by the Board of Supervisors, came within one vote yesterday of being abolished altogether.

In a 5-to-4 vote during a closed session, the supervisors extended the Park Authority's charter, which was to expire yesterday, until May 19.

If the charter had been allowed to lapse, the authority, which operates a network of more than 300 parks, would have ceased to exist. After the vote, the board voted publicly, 8 to 0, with one board member absent, to extend the charter.

The closeness of the first vote reflects county supervisors' deep dissatisfaction with the authority, which they have accused of mismanagement and slow work on new projects.

Some Park Authority board members, while acknowledging the agency has faced some management problems, contend that the county board merely wants to exert tighter control over the parks and use them for political advantage.

"I have trouble with any agency that doesn't answer to the people," said Supervisor Elaine McConnell (R-Springfield), who voted in private against extending the Park Authority's charter. "I don't see any accountability."

Voting with McConnell were Supervisors Nancy K. Falck (R-Dranesville), Martha Pennino (D-Centreville) and Joseph Alexander (D-Lee), according to board members. The board discussed the issue in private on the grounds that it was a legal matter, one of the provisions of the Virginia open-meeting law that allows elected officials to hold private meetings.

The Board of Supervisors has proposed a new agreement that strips the Park Authority of much of its independence and puts the parks under the control of County Executive J. Hamilton Lambert. The agreement would effectively place the Park Authority, which now enjoys considerable independence, in the status of most county agencies.

Last week, the Park Authority board accepted the agreement on a 5-to-3 vote, contingent on an opinion from Virginia Attorney General Mary Sue Terry that the takeover is legal. The Board of Supervisors voted yesterday to seek Terry's opinion on the issue.

Separately, Board of Supervisors Chairman John F. Herrity reopened the redistricting issue and threatened to call for changing the number of supervisors in an effort to bypass a new law that forbids the county to redraw the boundaries of its eight magisterial districts before the 1990 Census. The bill was passed by the Democratic-controlled General Assembly in an effort to prevent the Republican-controlled Fairfax board from redistricting at the expense of county Democrats.

Under the bill, however, Fairfax may redraw its boundaries before the November election for county supervisor if it changes the number of districts from eight to five, six, seven, nine, 10 or 11.

Herrity, calling the state legislation "the most preposterous state of affairs I have ever heard of in my life," said he planned to hold hearings on redistricting late this year or early in 1987.