The Justice Department, in an effort to break a stalemate that has threatened U.S.-Soviet relations for more than a year, yesterday filed suit to force the small community of Glen Cove, N.Y., to allow Soviet residents to use municipal recreational facilities.

In the spring of 1982, Glen Cove Mayor Alan M. Parente revoked all beach, tennis and golf permits issued to residents of Killenworth, a 36-acre estate owned since the early 1950s by the Soviet Union and maintained as a residence for Soviet representatives to the United Nations. Parente's action was later reaffirmed by a formal 6-to-1 vote of the Glen Cove City Council.

City officials were angry because diplomatic property is exempt from local property taxes under federal law. The officials decided that, because the Soviets were not paying property taxes in Glen Cove, they would not be allowed to use recreational facilities unless they paid a special fee. The Soviets refused.

J. Paul McGrath, head of the Justice Department's Civil Division, said yesterday that the lawsuit was filed in U.S. District Court in Brooklyn after more than a year of negotiations failed to persuade Glen Cove officials to change their minds.

The issue of beach and golf privileges in the Long Island community has been a serious one for the Soviets. On Aug. 5, 1982, in retaliation, the U.S. Embassy in Moscow was informed that until Glen Cove lifted its prohibition on Soviet use of local recreational facilities, U.S. diplomats and their families would be prohibited from using the "Diplomatic Beach" on the river at Nikolnaya Gora.

Later, the Soviets increased the rental fees for a Moscow athletic field that the U.S. Embassy used for softball and other sports. The fees were raised from a token amount to several thousand dollars.

The lawsuit contains a declaration from Thomas W. Simonds Jr., head of the State Department Office of Soviet Union Affairs, that "Should the Soviet Union proceed with its stated aim of taking further measures to deprive U.S. diplomats and their families . . . of recreational facilities, the matter of the use of recreational facilities could easily become a major political issue between the United States and the Soviet Union.

"A series of greater and greater reciprocal retaliations could be instituted by both sides, a course which would not be in the foreign policy interests of the United States," he said.

Arthur A. Hartman, U.S. ambassador to the Soviet Union, said that the Soviet retaliation has caused a "drop in morale among the staff and a consequent decrease in its ability to carry out our mission."

"We are still denied access to the Diplomatic Beach, and the Soviets have made access to other facilities difficult," Hartman said. "The few recreational facilities to which we have access are entirely under the control of the Soviet government, which can declare them off limits to us without consultations or advance notice."

The current dispute is not the first between Glen Cove and the Soviets. In the late 1960s, the city attempted to assess taxes on the property and scheduled a tax sale when the Soviets refused to pay. The Justice Department was forced to intervene, and in 1971 a federal judge decided that Killenworth was fully exempt from local taxes.