The faithful marched and sang again for the Rev. Martin Luther King Jr. today, this time as part of the establishment and with the police on their side. The occasion was a lively, sometimes unruly, "people's parade" in the city where he was born, nurtured and buried, an event that led a coast-to-coast outpouring of tributes to the martyred prophet of nonviolent revolution.

The day also brought renewed calls for an end to apartheid in South Africa, a cause that many think has revitalized the moribund civil rights movement in this country.

In Washington, where city officials have been honoring King on his birthday since 1969, thousands jammed the Convention Center and later the Kennedy Center to attend tributes. Local leaders took a special satisfaction in helping the nation's capital observe the federal holiday for the first time.

In Montgomery, Ala., about 500 blacks gathered on the steps of the state capitol to hear a proclamation from Gov. George C. Wallace (D) honoring the slain civil rights leader. The governor, who is recovering from surgery, did not attend. The proclamation was read by an aide to Wallace, but the moment was not without its irony: It was on those steps in 1963 that Wallace vowed "segregation now, segregation forever."

In Wisconsin it was a different scene, as volunteers led by Gov. Anthony S. Earl (D) helped snowbound farmers pick corn.

In Plaquemines Parish, La., all 485 students at a black high school stayed home to protest their school board's refusal to join the rest of the state in recognizing the holiday for King, who was born Jan. 15, 1929, and assassinated in Memphis on April 4, 1968.

In Buffalo, vandals whitewashed a bust of King in a city park Sunday night, but a parks crew cleaned it in time for today's celebration.

The day began here with a spirited countdown to midnight, punctuated with the singing of freedom songs, in a restored railroad depot built just after the Civil War. The unsung veterans of more painful marches had gathered, and disciples of King such as Marie Foster, a small woman from Selma, Ala., recalled the cattle prods and clubs.

"They told us blood would flow down Highway 80 if we marched," she said of the first, violence-ridden attempt in 1965 to march from Selma to Montgomery. "I bandaged my swollen knees, because I was beaten down on Bloody Sunday by state troopers."

Two weeks later, under the protection of federal troops, the marchers resumed. "We walked all 50 miles. It took five days," said Foster, who was wearing one of the orange canvas vests that she noted "were given to the people that walked every step of the way."

When the old-timers marched together again this afternoon, the only threat they faced was from the enthusiasm of the predominantly black crowds that lined the sidewalks of Peachtree Street and Auburn Avenue here.

So many people choked the street in front of the reviewing stand that they halted the progress of the parade, which ran nearly two hours late. Atlanta Mayor Andrew Young sat helplessly, smiling.

This morning, city, state and federal officials plus diplomats and champions of civil rights from other nations gathered here at Ebenezer Baptist Church, where King used to preach.

The Rev. Joseph E. Lowery, head of the Southern Christian Leadership Conference (SCLC), which King founded, issued the "call to commemoration" at the three-hour service. Referring to the Reagan administration as attempting to "change the cry of 'We shall overcome' to 'We shall overturn,' " Lowery said, "The truth of the matter is, in the name of Martin, we ain't going back.

"We've come too far, we've worked too strenuously, we've marched too long, we've prayed too hard, we've wept too bitterly, we've bled too profusely and we've died too young."

Vice President Bush nevertheless drew a warm response from the standing-room-only crowd, some of whom arrived hours early.

Bush praised King's philosophy of nonviolence and said the gathering of black elected officials before him was "colorful testimony" to its effectiveness.

On the subject of South Africa, Bush said, "In this sacred place, I call again for the end to apartheid." He said the administration intends "to remain involved" in pushing for change, and expressed the hope that "South African leaders will be able to find the same moral courage" as King and remain committed to nonviolence.

King's widow, Coretta Scott King, presented this year's Nonviolent Peace Prize to Bishop Desmond Tutu, one of numerous tributes paid to the South African in connection with the King holiday. As head of the King Center for Nonviolent Social Change here, King's widow had a strong hand in shaping the week's celebration.

Accepting the award, Tutu said of his black countrymen: "When we are free, we want to be able to say the leaders of the free world were on our side . . . . When we are free, we will remember who helped us."

Also addressing the crowd were Sens. Robert J. Dole (R-Kan.) and Edward M. Kennedy (D-Mass.), Mayor Young, Secretary of Housing and Urban Development Samuel R. Pierce Jr. and Georgia Gov. Joe Frank Harris.

In addition to being a new federal holiday, King's birthday is a legal holiday in 27 states, including three that honor Confederate generals the same day. Two other states, including Maryland, observed the holiday on the actual birth date, Jan. 15.

The celebration came at a time when many blacks say they believe the lode of white guilt over discrimination is mined out, and others are calling black middle-class guilt to account.

The official events surrounding King's birthday generally avoided controversial topics and focused on reconciliation and his legacy. But mingled with the harmonies of the old civil rights anthems, notes of discord and drift within the movement occasionally sounded.

Some civil rights leaders this week have voiced dissatisfaction, usually anonymously, with the style and strategies of Coretta Scott King.

Former King lieutenant Ralph David Abernathy, who celebrated the holiday in Alaska, complained that he was "left out" of arrangements here.

Atlanta City Councilman Hosea Williams, an advanceman for King who said he has been to jail "118 times," noted as the day began that "we voted white out and voted black in, and things really got worse for our people."

Williams and Abernathy became controversial in the black community by endorsing Ronald Reagan for president in 1980. Both have since had second thoughts, saying they were taken in by administration promises of more jobs for blacks.

"We once determined the political agenda," Williams said today to his former fellow marchers. "We can run America again, if we can only but come back together."