I have been making parachutes from umbrellas for years. The nylon is from an old umbrella and is perfect for them. Now that I have kids I get to use their old ones to make new toys for me and them.
I made this Dora parachute for my daughter. The weight/passenger came from the handle of the umbrella. My four year-old is not big enough to throw high enough, but she does enjoy trying to catch it.
Step 1: Skinning the umbrella
Start by cutting the fabric from the ribs of the broken umbrella. I use a box cutter or xacto to carefully the threads that the hold the material to the metal ribs. If you can salvage the tip leave them on, you can tie the strings to those.
Here are some that I have some saved up for a rainy day.