Not much adheres to nylon. Umbrella fabric with a tight weave also won't provide an opportunity for paint to get through pores and lock itself to the material. The weave will provide more surface area than a smooth film, but I wouldn't expect paint to stick long term unless you can find some designed for nylon.
There's something you could try (haven't tested it myself). Some hard-to-bond plastics become more active when they are heated. Heat is often part of the process for gluing them. Hot melt glue will often stick to some of these. You could try spreading a very thin layer of low-temperature hot melt glue on the area you want to paint. Then use a hot air gun at the low setting to melt the glue to a watery consistency, but not so hot that it scorches. It will self-level and flow into nooks and crannies. You want the thinnest film you can create, so put down very little glue and add to it if some areas don't get covered by the time you're done.
The area will be stiffer than the plain fabric, but a very thin film will still be very flexible and not prone to peeling off on its own. Paint will stick to the hot melt glue, and even if the glue doesn't make a "permanent" bond to the nylon, it will stick a lot better and longer than paint.