any of the products made for shower tile and grout? Bleach and alcohol will kill it when it's there, but can't really keep it from coming back. These sprays, though, are (allegedly) meant to keep it from coming back. You'd probably have to re-spray periodically--and experiment with the time intervals to see how long you can go.
I'm also a big believer in white vinegar and baking soda--I think they're the household gods of helpful hints. Maybe first try putting some white vinegar diluted with a little water in a spray bottle and spray it on the furniture occasionally. If the acidity doesn't discourage growth, then try making a solution with baking soda and water to see if the alkaline will discourage growth. The downside--the vinegar obviously has that distinctive "fragrance;" the baking soda solution may leave a fine powder residue after a while, which isn't a problem on the umbrella, but you're guests may object to its presence on their chairs. So maybe try the shower sprays first.