Pedant alert, but accurate use of language is important on the railway (and in many other places). And also hoping that I am up to date with the correct terms or, more accurately, they haven’t changed since my far off days.
The practice of leaving an item of rolling-stock shut-down between duties, maybe overnight or a weekend, maybe only a few hours, is called stabling. Multiple-units can go into stabling sidings, specially designated sidings primarily intended for stabling, or can be left on running lines, although this is never an ideal option.
Storing is putting an item of rolling-stock out of use for a lengthy period, probably for at least several months, and for a which a certain amount of preparatory work is carried out. There used to be two categories, Stored Serviceable and Stored Unserviceable, depending on the amount of work needed to bring the stock back into use.
Parking isn’t used, except in the sense of parking brake. Illogical, I know.
Berth is a bit of an old one, but I bet some old Southern Region men still use it for EMUs. Berthing sidings and carriage sidings probably still appear; the latter are not just for hauled carriages.
If someone has a better or more up-to-date definition you are welcome to give it. My experience is from the 1980s.