No, purchasing the original does not grant you the copyright. That stays with the artist unless they explicitly sell it.

Just to show you how untrue your statement is: if you get a tattoo on your body, you don't own any rights to that tattoo -- the tattoo artist does. The tattoo artist of Mike Tysons face tattoo receives royalties anytime it's shown in marketing or on TV.

If you are an actor, the tattoo artist must grant permission to the production for the tattoo to be displayed (otherwise it must be covered up.)