I know that beauty parlors rent out spaces to hairdressers. What do they call that space? Is it a space, a workspace, a chair, or something else? I need to know for a story I am writing.

They can rent a "booth" at the parlor.

The term is to rent a chair

Maybe it varies around the country. Locally the salons post signs booth rentals available. Of course this does include the chair.

I have always heard it refered to as renting a chair. --FLO--

Same here i have always known it as "Renting a Chair"

Looks like both are correct. My neighbor a cosmotologist calls it renting a booth.

http://www.beautytech.com/salonpub/booth.htm

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