Parades play tunes, and generally there are several sets of words to good tunes. Some of them are going to be sectarian, some less so, or not even so at all. The words people hear in their heads may be different from the words the organisers say they were intended to be hearing.

To take examples from other contexts. So there you are playing "Maryland my Maryland" in a picket line, and get accused of playing "The Red Flag". There's a hymn that uses the same tune as the German National Anthem. That would really have gone down well in the war...