The unit of scale is the function app for the host instance. So, when a trigger is invoked, the scale controller will load the function app and cold start the triggered function. The scale controller is important because it determines how many host instances to allocate.
Referring to Service limits, a consumption plan can receive up to 100 ACU for a consumption plan. This is the "server" mentioned in Steps 1 and 2 under What happens when I write a function on https://azure.microsoft.com/en-us/blog/understanding-serverless-cold-start/ but you are correct in that each host instance gets 1 CPU. But I can't say that parallelism can't be done. It depends on the underlying hardware and runtime you're using, which I don't know the absolute specifics for a consumption plan. What I can say and suggest though, is that if you're trigger function is lacking the necessary CPU performance to quickly execute, then scale up to a higher plan. Before doing that, make certain you leverage detectors like Functions Scale Overview to better understand how your app is scaling and where you can make optimizations.