I'd argue, parody has gotten far more popular, though far worse than it was in the past. The question wording honestly should be answered, No, in fact far more so than in the past. Comedy has died in most movies and most shows. But that doesn't take out the fact its still a popular genre.
Parody is a timeless form of comedy. Weird Al is a very popular maybe iconic artist but he is far from the first to create parody songs. Parody has always been enjoyed as a fun musical niche. People enjoy laughing and parodies are an excellent way to connect with any audience by pocking fun at whatever is popular at the time.
Parody songs may not be heard as often as they once were but I do believe that they will make just as much of a comeback at Weird Al. Recent social media would suggest that people are very excited for the come back of Weird Al, and as a result are looking forward to the parody songs that he is so well known for.
Parody songs, like the ones popularized by Weird Al, are due for a comeback. They were a fad, similar to bell bottoms, that disappeared for a long time. New generations of younger music listeners will appreciate the clever comedy and musicality of parody music since they haven't previously experienced it.
I believe they are just as popular as when he first started, because now with You Tube and other social networking sites, anyone can do a parody and post a video. Look at what happened with that song during the Olympics. I think it will be hard for Weird Al to compete with the millions of creative people online.