How to install Python on macOS Catalina using homebrew?
chown the usr/local directory:

sudo chown -R $(whoami) /usr/local


Then try to install python3 again and it should work:

brew install python3


OR,

You can try this also if the above solution didn't help 

sudo chown -R $(whoami) $(brew --prefix)/*


Reference: https://gist.github.com/irazasyed/7732946