Install the current version of mysql.

    # Install current mysql version
    brew install mysql
    
    # Start current version of mysql (including on login)
    ln -sfv /usr/local/opt/mysql/*.plist ~/Library/LaunchAgents
    launchctl load ~/Library/LaunchAgents/homebrew.mxcl.mysql.plist
      
    # Check current mysql version
    ls -la /usr/local/Cellar/mysql
      
Install the older version of mysql.      
      
    # Find older mysql versions
    brew search mysql  
      
    # Install older mysql version
    brew install homebrew/versions/mysql56
    
    # Start older version of mysql (including on login)
    ln -sfv /usr/local/opt/mysql56/*.plist ~/Library/LaunchAgents
    launchctl load ~/Library/LaunchAgents/homebrew.mxcl.mysql56.plist
    
Then to switch versions.    
    
    # Unlink current mysql version
    brew unlink mysql 
    
    # Link the older version
    brew link mysql56
    
Or force the older link (avoid if possible).    
    
    # To force the link and overwrite all conflicting files
    brew link --overwrite mysql56
    
Or use the switch versions command.    
    
    # To switch versions
    brew switch mysql56 5.6.27
    
To double check which version you're on.    
    
    # Check which version of mysql is currently symlinked
    ls -la /usr/local/bin/mysql
    
And if you only want one version running at a time.    
    
    # Stop current version of mysql
    launchctl unload ~/Library/LaunchAgents/homebrew.mxcl.mysql.plist
    rm ~/Library/LaunchAgents/homebrew.mxcl.mysql.plist
    
    # Stop older version of mysql
    launchctl unload ~/Library/LaunchAgents/homebrew.mxcl.mysql56.plist
    rm ~/Library/LaunchAgents/homebrew.mxcl.mysql56.plist