Parade routes always seem to be changing, but for many years it was 5th Ave that was seeing all the celebratory marching action. After the avenue became so popular, however, the city adopted a new policy and refused all new parade requests.
According to the NY Times, "on Tuesday, the United States Court of Appeals for the Second Circuit upheld a lower court’s ruling that the 'Fifth Avenue rule' did not amount to a restriction of free speech." The ruling is attached to a case dating back to 2005, when an antiwar group sued the city after not getting a permit for a protest march — they charged that cultural parades were favored over political ones.
It was back in 1971 that the city took on the Fifth Avenue rule, only allowing 15 parades a year — but in 2001 they "quietly adopted the ban into administrative code. No other street in New York has such a restriction."