My instinct is that I'd like to see all of Edinburgh's citizens attend this parade and turn their backs on it, silently.
I'm not sure in my mind if ignoring it by not turning up or ignoring it by turning up and turning away is more effective.
I accidentally stumbled out of Central Station and into an orange walk a few weeks back. While there was much triumphalism and banging of drums and waving of banners, there was a very thin crowd in attendance and most of that was following alongside the march. Like many such orders/organisations/religious bodies from yore, that is primarily pre-occupied with the preservation of yore, the LOL is one in a long, slow and possibly terminal decline, but which continues to dress the windows, stick its finders in its ears, pretend there's nothing wrong and go about its business as usual. Even if no-body is really turning up to watch.
A quick google of the BBC reporting suggests that in 2006, 18,000 bandmembers and walkers took part in the "Glorious 12th of July" in Glasgow. By 2009 that was 8,000 and this year it was 4,500 (and 4,000 spectators). I don't know the membership figures of the LOL (and I'm pretty sure that such a closed order keeps such things close to its chest), but they will have to pull out a lot of stops to get a parade of 20,000. (rumours abound on tinternets of bands being brought over from Northern Ireland).
They might all have to buy new outfits for their bands though if militaristic uniforms have been prohibited.