Role of mode in bottleneck function in unet network
That mode parameter is simply referring to how (what way) the bottleneck should be, what type of bottleneck to use.

There is nothing fancy to it. There is at least one usage in statistics of mode that I know of, as a way to describe some distributions, but this doesn't have anything to do with it.

As to the difference between cascade and parallel, in the former the layers pass the input to each other (like in the diagram), while in the latter the layers operate on the same input.

In general, cascade refers to the application of operations sequentially (e.g. in a cascade classifier). Parallel means that operations are performed independently and could, theoretically, be ran simultaneously (think about parallel operations for threads). Here, those layers are parallel since they all operate on the same input, and their output is then collected by subsequent layers.