do all the branches of run parallel action need to be completed before the next action can be triggered?
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Everything inside the parallel action will execute before any actions following it. (Just as a technical aside: this does not necessarily mean that everything will complete before moving onto the action following the parallel action: ie a command is issued to a server somewhere such as an update to an item. The update action will complete before moving forward, but its the server's job to make sure the item is actually updated here). You could think of everything inside the parallel action as a single action (just for logical sake) so yes they all execute before the next in line action.