I think your issue is that the dialog is modal. The log provided is not complete enough to speculate, however I have seen this before. Typically I would perform the click that initiates the dialog on one thread and then wait for create and dismiss the dialog on another as you have done. Do you ever see Web_ match in the log?
One thing you can try is to use a Signal component. Have the Web_.Created event dismiss the dialog (by clicking btnOK) and then call the SignalAll method of a Signal component you add to your Global Container. In your automation, after you call PerformClick on button1 call the Wait method of the Signal. If that returns true, then you can continue with the PerformDoubleClick method (the one you call after btnOK.PerformClick in your screenshot).
On success case robotics runtime log show Matching case, but Error case Matching log was not displayed.
Please refer the attached "Popup can not clicked.xlsx".
I don't know why Matching log wasn't displayed on Error case.
The partial logs provided here aren't really helpful unless you post more of them along with the adapter (either of which may contain sensitive information, so you should be careful to remove that).
I would suggest you open an SR to get support to assist you with this specific issue. I do believe my suggestions will work though, as I have used them before for a similar popup. If you do open an SR with support, please post the number here, so the moderators can track it.
You should perform the wait off of the parallel process instead of after it, however I do not think that is your issue. I think you should open an SR to have one of our support team look at this on your machine. I suspect it would be easy to determine the issue there.
If you do open an SR, please record the number here for tracking and resolution purposes.
You should be careful not to try to work with a UI on two threads. All calls will be marshalled to the application's UI thread and you could get into a race condition where one thread blocks the other. I would not use a parallel process here but instead a WaitForEvent.