Create your LED Strand
This parasol will need to open and close hundreds of times. A traditional LED strand won't stand up to this; LED strands are just not made to stretch and flex the way an umbrella does.
Using individual neopixels soldered together into a long strand works much beter and gives you more control over the spacing between lights. I like to use flexible stranded silicone wire to string them together. This wire will keep your parasol working for years even with lots of use.
Here's a video showing my favorite method of quickly stringing neopixels together.
Measure around the circumference of your parasol and decide what spacing you'd like for your lights. Mark and strip your wires, then slide the neopixels on and solder them in place. Finally, connect all the data lines between the neopixels, as shown in the video, making sure you're following the directional arrows on the pixels.
Once you've got all the pixels strung together, hook them up to a microcontroller running the Neopixel Strandtest code to make sure everything is working.
Plug your male JST connector into the female plug coming from your battery and take careful note of which side is power and which is ground.
Slide a small piece of heat shrink onto one of the power and one of the ground wires you just attached to your LED strand. Carefully solder each of these wires onto the JST connector and cover with the heat shrink.
Now we're going to add hardcore strain relief to this connector. It's the piece that takes the most tugging and wear and tear, so we want to be sure it won't break while you're changing batteries (in the dark, in the middle of the night, most likely).
Slide a large 1/2" or 3/8" piece of heat shrink over the connector, leaving the front open but covering the back and the connections you just made completely. Don't heat it yet!
Take a hot glue gun and fill the back side of the heat shrink with hot glue. Then, while your glue is still liquid, hit the whole assembly with a heat gun to shrink the heat shrink in a controlled manner. Be careful not to get glue inside the connector itself.
Once this sets up, your connector will be very tug-proof and will hold up beautifully to night time battery fumbling.
Connect the red wire from your power connector to one of the leads on your power switch. Connect another red wire to the other lead.
Splice two wires each onto the ground wire (coming from the power connector) and the red wire (coming from the power switch). Connect one of these wires to the neopixel strand + and -, and the other to Vin and G on the Teensy.
Plug your battery in and watch the lights twinkle.