Do you suffer the same competitive instincts with yourself, how do you deal with those? Unless you very narrowly scope the competition, you can never win - you might be the best 100m runner in the world, but you'll be beaten over 400m let alone a mile, or a marathon. So somehow you must rationalise your own life in the same way - you don't expect to win everything, even if you're a competitive personality.
Competing at primary school, where you may be comparing a few random collection of kids is madness, it's almost impossible to get an accurate view of achievement, and even then you don't know effort - 'cos the motivation side is normally hidden. Even if simple things like that 100m - was the kid who came second actually trying, were they tired from the 4 hours of gymnastics they'd done the night before etc. But really you know nothing, proxy competition in primary schools is so foolish.
I'm not actually sure that the "competitive parents" are always competitive - someone commented that one of the parents at DD's school was very competitive, but in all my conversations with them they'd not seem competitive just very unsure of themselves and the questioning and comparison came from a place of nervousness about where their kid was and wanting re-assurance that it was okay. Despite the kid achieving well.
That may of course have been their dynamic with me (where DD wasn't a competition for whatever reason.)
The viewpoint where you're disappointed because of the achievement of another is not good, as you say the rational side of you knows it makes no sense. Even without the comparison, you don't know if your child actually cared themselves - maybe they don't see the value in competing at the St Agatha's under 7's egg and spoon obstacle race, so phoned it in. That isn't necessarily bad, one of the important skills kids sometimes don't learn until too late is when to put the effort in - you can't always "do your best".
You do need to know when you should put the effort in, and also find things where you want to put the effort in of course.