Is parameterized complexity going to be the future of complexity theory?
Predicting the future is nigh impossible, especially so for cutting-edge research. I don't think anyone predicted how much impact deep learning is now having or that cryptography would be taken over by indistinguishability obfuscation.

That said, I will say this much: I don't see any particular reason to expect parameterized complexity to take over. It's a mature field that has been active for something like 20 years. It doesn't really strike me as an up-and-coming area. To be clear, I think it's a successful area that will continue to thrive.

If you look on google trends, search interest in parameterized complexity has been declining. (Stick in some other terms for a comparison if you're interested.) If you look up the combined citations for the Downey-Fellows textbook Parameterized Complexity and their updated textbook, you see that they are pretty stable:

(Source: Google scholar. I added both books to my own profile, merged them, took a screenshot of the combined citations, and then deleted them from my profile.)

This is a healthy number of citations, but it is not the exponential growth that would make you think parameterized complexity is going to take over. Of course, this data is very flawed, but it's the best indication I can find of the global popularity of parameterized complexity.

Note that things can be very popular locally even if they aren't popular globally.
When I was an undergrad, I thought that I needed to learn about category theory because everyone around me was talking about it; I even bought a book. Then I moved on to grad school and never heard about it again; the book remains unread to this day. 
Perhaps you are in a similar situation -- you are in a department where there is a lot of parameterized complexity going on, but, if you move somewhere else, the story will be completely different.