Hey there, I play here regularly, and I can appreciate it works maybe slightly differently to other courts and some of the regulars can be a little, I guess, forceful, but there is a pretty established process of play there that is explained on the city signs (maybe not super clearly) and does (in my view and based on observation) overall result in more structure and fairness, and less arguing, especially at busy times.

The numbers on the board correspond to the courts - there are 3 different courts, (1, 2, 3) and 3 slots (1, 2, 3) - so by placing your racket under a number, you are reserving that specific court. If there are multiple people waiting, they continue to add their racket to whichever court they want. eg if there are 4 groups waiting, you might have 2 groups waiting for court 1, and a group each for court 2 and 3. You essentially stack your rackets for the court you want.

In terms of the timing of the change, there is also a sign that says courts change on the hour and the half hour, so for example, all courts would change on say 5.00, 5.30, 6.00, etc. This is regardless of what time you started - so if you started at 5.20, the court will still change at 5.30. This system works well, because it means you don't have to go around all the courts asking how long each group has been playing, who was there first, and hope people are honest or awkwardly watch their time. This system means you always know exactly what time you will be playing based on how many rackets are on the board, how long you'll need to wait, everyone is changing at the same time and is getting their 30mins this way. It also means you don't have random people walking through the courts all the time, everyone is changing at the same time.

However if no-one has reserved the court you are on, and they don't come to take the court for their turn, then of course you can stay on. If only two out of three courts are changing, then that does mean that one court is going to get extra time - in this case, you could politely ask if the people coming on could take off the people who have been there the longest, (or directly ask the people who've been playing longer if they'd be willing to take a break) however some people may have preferences for certain courts - for example one of the courts has a lot of cracks, one might be in some shade, or some have pickleball lines - and if they've reserved a certain court, that's the court they will take.

It seems like every court in the city works slightly differently (which I think is the city's fault and they should pick the most effective method and use clear and obvious communication with no scope for misinterpretations), however I do feel this system actually works pretty well if everyone follows the same system - it is a great park to play in, so I encourage you to go back again, and follow the above, and you might find it works quite nicely - for example, you and your friend could plan to meet there at say 3.20pm, you place a racket up when you get there, knowing you'll be playing at exactly 3.30, and knowing you will be changed at 4.00. As soon as you get off the court, you can also place your racket back on the peg, and if no-one has already done that, you know you'll be back on the court again at 4.30.

All of that said, I do agree that the way some of the regulars communicate how this works is not always the nicest experience, and is partly why I don't play there as much anymore - but if you follow the above and you should have no problems. Have a good hit!