What paints could I use on nylon fabric?
I don't have specific experience painting nylon fabric, but as a starting place for you, Dharma Trading Company has a tutorial for painting and/or dying silk that may apply to your project. They sell several lines of fabric paints and dyes. Dick Blick has silk/fabric paints and dyes. There are videos on YouTube for painting fabrics. Go to the manufacturer's home pages to check for compatibility with nylon and don't hesitate to ask for their suggestions. Be aware that many fabric paints require being setting with heat, like an iron.

As an alternative, consider going that retro route and using painted and/or dyed silk. It can be exquisitely beautiful. Waxed or oiled silk, lacquered papers, leather, feathers, and palm leaves have all been used by various cultures over time for their umbrellas.