(CNN) -- On a day when New York City threw a parade to celebrate the New York Giants for their Super Bowl victory, at least one veterans' group is using the occasion to ask: What about a parade to honor the soldiers who fought in Iraq?
"If the Giants deserve a Super Bowl [parade], so do the 1 million Iraq veterans who have served," says Paul Reickhoff, an Iraq war veteran who now heads Iraq and Afghanistan Veterans of America. "The American people want a chance to say thank you, and want a chance to respect and remember those who served."
Along the parade route Tuesday in New York, several parade-watchers voiced support for the idea.
"Without a doubt there should be one," said Jim Lynch, who said his brother Scott was killed in Afghanistan. "They deserve it."
"Amen. Absolutely," said his friend Jim Diffley. "We showed up at 7 a.m. for a Giants parade; I'd be here at 4 o'clock yesterday for an Iraq war veteran parade."
Some of the Republican presidential candidates have already picked up the issue. When he was a candidate, Gov. Rick Perry of Texas drew sustained applause when he criticized the administration for failing to sponsor a parade, and former Gov. Mitt Romney of Massachusetts has also weighed in.
"Whether that's a ticker tape parade, or whether it's a special ceremony somewhere else, welcoming our troops home is something they deserve," Romney said recently on Fox News Channel.
But senior Pentagon officials said the time is not yet right for a high-profile national event, because U.S. soldiers are still fighting in Afghanistan.
According to Col. David Lapan, spokesman for the chairman of the Joint Chiefs of Staff, "The chairman and other senior military leaders do not think it's appropriate at the given time, with service members in large numbers fighting in Afghanistan, to hold a national level parade in New York City."
Lynch disagreed with the suggestion that a parade might be premature.
"What's too early? How many people gotta die before it's too early?"
In St. Louis, Missouri, organizers were not going to wait. They threw a parade honoring Iraq veterans on January 28 that drew thousands, and other cities may follow suit.
Darrell West of the Brookings Institution said the argument shows how controversial the Iraq war remains.
"Even when it's over, people are debating whether we won or lost," he said. "If you put on a parade, or have some other celebration, people are going to interpret this as a victory celebration, and the American public still remains very divided over what happened in Iraq."
On Monday, the White House announced that President Obama and first lady Michelle Obama will host a dinner later this month with some Iraq veterans and their families to honor those who served in Iraq. The event is described as "an expression of the nation's gratitude for the achievements and enormous sacrifices of the brave Americans who served in the Iraq War."
CNN's Richard Roth, Meghan Rafferty, and Larry Shaughnessy contributed to this report