Open your parasol and remove any cats.
Glue or stitch the teensy next to (not on top of) one of the parasol's spokes.
Set the LEDs facing inwards along the edge of your parasol. You can use glue or a needle and thread to secure them. Hot glue works great for this, and goes on quickly, but may not last as long or be as robust as a good fabric embellishing glue.
Or, you can take the time to hand-stitch the LEDs to the parasol using a needle and thread. This takes a bit longer but will very likely hold up the best in the long run.
Take a small piece of netting or fabric and make a pocket for the battery along one of the parasol's spokes. I like using netting since I can poke the connector through one of the netting's holes and it'll stay neatly in place.
Open and close your parasol a few times to make sure everything works and stays put.
Take note: This parasol is NOT WATERPROOF. If you are adding these LEDs to a true umbrella instead of a lacy parasol, you may want to add some plastic or waterproof fabric to your electronics so you can use it in the rain.
If you want, finish up by gluing or sewing lace trim over the top of the LED strand to hide the wiring. The lace will diffuse the lights and give a softer and more finished look. It also makes the parasol look a bit brighter from the back since some of the light is being reflected back onto the parasol's top.