You are so right that so many people just don't understand the way their dogs' minds work and in one way that will always make dog parks an unpredictable place to take your dogs.
It is very interesting that you noticed many squabbles between the dogs on the same day in the same area.
This is something that everyone using dog parks, or places where there tends to be a lot of dogs, to excersise their animals should think about.
I am in the middle of research at the moment to investigate why this happens. I have not concluded the research and the science is still a bit rough around the edges but you may be interested in my findings so far....
It seems that dogs hearing other dogs fight, even if it is out of human hearing, become more defensive - probably in order to protect themselves and prepare for battle if the fight comes their way. Marking behaviour becomes more frequent in these cases and it seems likely that the chemistry of the urine left is altered to signify the readiness to defend, thus informing more dogs that there is trouble afoot. With just these two theories it is obvious how a ripple effect can happen within a park containing many dogs and how the rate of fights can escalate beyond the odd tiff you would expect.
The other factor I am considering is the change in weather - at this time of year wild northern hemisphere canids are starting to pack more tightly. It seems sensible that older Alpha animals begin to feel the cold and the effort of keeping warm as well as having energy to hunt will tell on them more than a younger animal. This may be the opportunity that younger dominant animals have been waiting for to challenge, seriously, for leadership. This may be the trigger for some of the fights that are occuring in established dog parks. It is also the time of year when most two or three year old animals are getting comfy in their adult skin and feeling pretty confident - this may also be a factor as most puppies (in the wild, at least) are born early in the spring and will reach the peak of fitness and confidence at this time of year. In pet dogs this may be a residual behaviour, not actually needed but still somewhat in evidence.

As I say - this is pure theory at the moment. There does seem to be a change in a pet dog's sense of well being at this time of year - I have questioned and visited 27 families so far that have watched and recorded their dog's behaviour over the exceptionally hot, dry summer we have had. We are still very dry - but the nights are drawing in and it is certainly colder. I have asked them to monitor their dogs until next summer as I am theorising that the same upsurge will happen in the spring.

We shall see....

Very interesting anyway.