I think the bigger issue here is that you are obviously uncomfortable with the idea of approaching people in public and your parents are treating this as irrelevant and something you are supposed to just force yourself to do it anyway despite feeling like the situation is wrong and threatening. You shouldn't need to justify not wanting to do that by appealing to some kind of cultural authority about what is acceptable to society.

Personally even as a man it normally freaks me out when strangers approach me in public. It just feels like a very unusual, unexpected and potentially unsafe kind of circumstance, almost never something positive, there's no way I would trust such a person, so I'm not going to do that to others because it's like I would be inflicting that on both of us simultaneously, and that would of course come through in any interaction I attempted. How could I expect them to be receptive to that when I would never be myself? People may argue, that's the wrong way to feel and so it doesn't matter, replace that attitude with a better one, as if they themselves could easily substitute a totally different way of being for how they are.

If you need an invitation in order to feel safe in a social situation, I would say it is ok to demand that people respect that and not mock you for it.