Connect to streaming Twitter API, parse tweets, and write to CSV in real-time
MyListener is not a very descriptive name. Maybe CSVWriter would be a better one. And that name already suggests something else, that it should take the file name as a parameter:

class CSVWriter(StreamListener):
    def __init__(self, file_name, *args, **kwargs):
        self.file_name = file_name
        StreamListener.__init__(*args, **kwargs)

    def on_data(self, data):
        try:
            tweet = parse_tweet(data)
            content = extract_content(tweet)
            with open(self.file_name, 'a') as f:
                writer = csv.writer(f, quotechar = '"')
                writer.writerow(content)
        except Exception as e:
            logger.warning(e)
        return True


It is also enough to except Exception. BaseException will catch a bit more than you would probably want, namely KeyboardInterrupt (so you can press Ctrl-C to stop the program).
        ...

twitter_stream = Stream(auth, CSVWriter("tweets.py"))


There seems to be no need to first save tweet['CREATED_AT'] = parser.parse(tweet['created_at']) and then later do tweet['CREATED_AT'].strftime('%Y-%m-%d %H:%M:%S'). Just do the correct conversion at the end (I don't know which one exactly, because I don't know what format it is in, orginally).

I would also reduce the number of blank lines. While some are nice for readability (to separate blocks of code) or are recommended by PEP8, blank lines between before elif or else block just break up the reading and reduce readability.