Right after you enter the door for Bowser's stage, you need to press space before the fade-out is finished. This will allow you to skip over the Bowser's fight, ending the game faster. I found more success after spamming space when I see the start of the fade-out.

If it isn't working, close Flash Player and restart the game that way. It seems to only work the first time you run through the game. When you try it again without closing it, it just doesn't work anymore.