Here's a post that might help you - https://www.homestoriesatoz.com/diy/how-to-paint-a-rug.html
You can only really paint a flat pile rug. Use a roller - that's best!
There are lots of ideas here:
If flat pile, with a roller and Chalk Paint. When dry seal with Clear or antique wax.
Here is a video that will help: https://www.youtube.com/watch?v=aghhzJMscZs