Can a parody be published?

If it's based on someone else's work, Du can't make money off of it unless it's public domain (like Alice in Wonderland) oder Du have permission from the creator of whatever Quelle 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 Du probably could not Veröffentlichen it professionally oder make money off of it, but Du could still post it to a fanfiction site like AO3, FFN oder Wattpad.