Get an old umbrella with a good sturdy structure. The fabric part of it only has to be in quasi good shape (that's your pattern) make sure it's one of the ones with the little metal caps on the end that come off. Make sure all of those are still attached too.

Take the fabric part off. Try not to tear it. Take it off gently, and cut small slits in it around the top part, if necessary to get it off. Carefully get the little metal cap things off and put them aside. You want to be able to use them later.

Pin the old top to the fabric (use something that isn't too stretchy, or not at all). Cut the new top out carefully, leaving at least an inch around the edge for the "hem." Where the center is, cut out the absolute smallest hole necessary to get the top thingy through, too small is better than too big.

Hem the edge so it will fit. Put it through the center thingy, sew it in if necessary (this will be a little tricky). Clean all the old fabric and glue out of the end caps (make sure your fabric isn't so thick they won't fit on) and presto. If you were really careful with your pining and cutting and hemming (oh my!), you should have a parasol.