But you do not necessarily have to be competitive to become the team leader or organiser, simply being seen to be good at something, like leading or making decisions.
I was appointed to managerial jobs at work twice because there was a problem in the department and I was considered to be the person most likely to deal with it effectively.
In each case, I applied for a job, so was inevitably in competition with others chosen for interview, although I had, and still have, no idea who they were.
But competitive in the sense of boasting about my children's achievements, no, that is not my scene. I suppose I do not look at the right social media, but it happened even before social media.
Actually, parents who have children who are doing things exceptionally well, exceptionally young, uually keep very quiet about it.