Council Speaker Christine Quinn, one of the state's most prominent Irish-American elected officials and a front-running mayoral candidate, again plans not to march in the Fifth Avenue parade, held this year on March 16.
But gay people do march in the parade and are welcome, said Hilary Beirne, the parade's executive secretary.
Dole kicked off his day in Wheaton, Illinois, where he was to march in a parade and attended a picnic.
The main candidates seeking the Republican nomination, including front-runner Joe Lhota, a former chairman of the Metropolitan Transportation Authority, each said they plan to march in the parade.
Irish gay advocates sued in the early 1990s, after parade organizers refused to let them march with an identifying banner.
About 1, 200 people watched soldiers from 1st Battalion The Royal Irish Regiment march on to the parade square at Clive Barracks, Ternhill.
Former X Factor singer Amelia Lily is headlining a free concert at Custom House Square on Sunday 17 March, with a parade also taking place in the city centre.
New Orleans "Indians" are black people who march in the Mardi Gras parade, wearing 100lb (45kg) dresses that they sew themselves throughout the year.
The centennial celebration will continue with events throughout 2013 including a performance piece by artist Nick Cave March 25-31 and a parade of historic trains May 10-12.
The city of Beverly called off its parade because so few veterans would be able to march.
Earlier, Mr Osborne said it was illegal to organise or participate in a parade without prior notification and said stopping such a march was one option open to police.
For the first time, up to 8, 000 visitors from around the world were due to march in a "People's Parade" on Sunday, when Ireland's capital city also intends hold its usual procession of bands and pageantry.
The Irish government has told hauliers they will need special permits to move animals from any farm, more soldiers have been called in to guard the border, and this month's St Patrick's Day parade in Dublin, due to take place on March 17, has also been called off.
The parade's backers say gay people are welcome to march as long as they abide by the rules.
Sinn Fein councillor Monica Digney said she was "totally opposed" to the parade taking place and criticised the Parades Commission for giving the march the go-ahead.
BBC: NEWS | UK | Northern Ireland | Police praise after town parade
The relatives had challenged the commission's ruling not to impose any restrictions on a nationalist parade through the town on St Patrick's Day, 17 March.
Mr Lazio's failure to march alongside Hillary and Rudy in the gay-pride parade may have cost him some homosexual votes, despite his claim that only previous engagements kept him away.
The speaker said she hopes the parade organizers will change the policy soon, especially because she would love to march with her father, who is 86 years old.
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