In Australia, we don't really have 'flag etiquette'. I mean, there are military rules about the flag, but most people don't know them and wouldn't be expected to abide by them (only very simple ones, like you shouldn't hang one upside down, and it shouldn't touch the floor). I can't even think of a time when I've seen the Australian flag 'paraded around', and I would have no idea what to do if that did happen! While we are taught to stand and be quiet during the national anthem (unless we are singing it of course), I have never seen, nor would expect to see anyone, placing their hand on their heart or anything like that. The most frequent occasion where the national anthem is played is for major sporting events, and while most people stand, I haven't seen many remove their hats. It's a very different culture.
While I'm travelling, I try my best to be respectful to the different cultures that I come across. In Asia, that means covering my shoulders and knees in sacred places (temples etc), and generally dressing slightly more conservatively. In the US, it means being more mindful of what I say (a lot of words that are perfectly acceptable to use in Australia are considered cursing in the US), and of course standing and being quiet for things like the flag retreat ceremony and national anthem. If an American flag was 'paraded' in front of me, I honestly wouldn't know that it was a 'thing' when people are expected to show a particular type of behaviour.
In future trips to the US, I will definitely keep this in mind, and make sure that I stand in silence if I see a flag being 'paraded'. That is all I would do for my own country's national anthem - I would feel weird doing any more than that for some other country's anthem/flag.