Can a parody be published?

If it's based on someone else's work, you can't make money off of it unless it's public domain (like Alice in Wonderland) or you have permission from the creator of whatever source you're using (typically this means paying for the rights to use the material, if they're even willing to sell).

So in answer to your question, no you probably could not publish it professionally or make money off of it, but you could still post it to a fanfiction site like AO3, FFN or Wattpad.