From the point of view of the 1st Amendment rights to freedom of speech and peaceable assembly, I think they should be treated as equivalent. Both are political speech, and the government must not favor pro-status-quo speech (like a 4th of July parade) over anti-status-quo speech (like a protest march).
Indeed, if they're going to favor one, I'd rather it be the anti-status-quo speech. The US is founded on the notion that we are always seeking a more perfect union. The reason the Great Seal [1] has an unfinished pyramid because we should never think we are done.
I don't disagree with this, it's actually a portion of the point I was trying to make. That is, all other things equal, they should be equivalent. It's the "other things" here which may make the results different, and I'm not sure that's a bad thing.
For example, consider a parade scheduled from 11 AM to 1 PM on Sunday, known well in advance, and an impromptu protest that really gets way at some random point in the afternoon, most people didn't know about, and causes logistical problems for many people. I value and appreciate the need for both, but I also understand and support the police trying to contain and in some cases disperse (peacefully!) the second. Indeed, the anti-establishment bent the protests often have usually benefits from some police presence, IMO. It's a better story and reaches more ears.
Indeed, if they're going to favor one, I'd rather it be the anti-status-quo speech. The US is founded on the notion that we are always seeking a more perfect union. The reason the Great Seal [1] has an unfinished pyramid because we should never think we are done.
[1] http://en.wikipedia.org/wiki/Great_Seal_of_the_United_States