Government prosecutors yesterday asked the U.S. Court of Appeals here to allow them to use evidence that they say is essential to prove their contention that John W. Hinckley Jr. was sane when he shot President Reagan.

In papers filed yesterday, the prosecutors asked the appeals court to overturn a Nov. 17 ruling by U.S. District Judge Barrington D. Parker that law enforcement officials violated Hinckley's constitutional rights last March 30 when they questioned him without a lawyer present shortly after his arrest and later when they seized papers from his prison cell. Parker ruled that, as a result, Hinckley's statements and papers cannot be used as evidence during his trial.

Since Hinckley's lawyers have conceded that their client shot Reagan and three other persons, the only issue at his trial will be his mental state at the time. Hinckley's lawyers have said they will argue that he was insane at the time of the incident and cannot be held criminally responsible for his acts.

Although a battery of psychiatrists examined Hinckley in the weeks following his arrest, prosecutors believe the testimony of law enforcement officials who talked with him just after the shooting will be vital to convincing a jury he was sane at that time.

In his ruling, Parker said that FBI agents, by questioning Hinckley after he requested to see a lawyer, violated safeguards established in the famous 1966 U.S. Supreme Court case Miranda v. Arizona, which requires police to inform suspects of their rights to remain silent or to have a lawyer present during questioning.

Parker also said that federal prison officials violated Hinckley's right to privacy when they seized personal notes, which reportedly included a fictional conspiracy plot contrived by Hinckley from his cell at the federal correctional institution at Butner, N.C.

In their appeal, the prosecutors argued that "virtually all of the questions asked by FBI agents were "standard biographical questions which the FBI routinely asks . . . and were not intended" to get incriminating evidence from him.

Prison officials did not violate Hinckley's rights when they seized his notes and a diary from his cell, the prosecutors argued, because they were reviewing his writings for "indications of his state of mind" in light of a "serious risk of suicide." Hinckley has attempted suicide twice since his arrest.

Hinckley's lawyers argued that Parker had interpreted the law correctly in ruling that the statements could not be used at trial. The seizure by prison officials of Hinckley's papers was "at best an 'exaggerated response,' " the defense lawyers said, quoting Parker, to concerns over a possible suicide.

The battle over admitting the evidence, which has delayed Hinckley's trial, is expected to be heard by a panel of appeals court judges next month.