Yes but with complexities in general, and a collision with non-copyright issues for deep fakes that starts to get into more unexplored societal/legal territory. First, parody is part of Fair Use, which means that it's an "affirmative defense": in a lawsuit, the burden is on the defendant to bring it up and prove it. That's in contrast to ordinary defenses or arguments around the facts asserted by the plaintiff, where it's up to the plaintiff to prove them to whatever the required standard of evidence is. In practice, that can mean a somewhat higher financial risk and higher chance of losing at the edges.
Second though, parody (and Fair Use) is about copyright and trademark, protecting use of such material for commentary and so on. But use of someone's likeness directly, particularly for someone famous, in order to produce new works is arguably something new that hasn't really been dealt with yet. Jay-Z making a copyright claim definitely seems dubious, and perhaps was done simply for convenience rather then legal strength, copyright disputes are the form in which most take down systems work. I can see arguments both ways for whether copyright would apply at all: in favor, the argument would be that the ML models are being trained via copyrighted works, which in turn makes them derivatives. On the other hand, facts are not copyrightable (in the USA) regardless of effort or source. A counter argument would be that the ML models are merely deriving facts about a person's vocal cord, facial structures and other physical natural characteristics, which then create a factual physical model which can be utilized to produce new works. In that case, all these new deep fakes would be their own brand new copyright (and potentially the ML models themselves not copyrightable). That'd be an interesting legal argument to see hashed out.
But even if they're new copyrights, right to voice & likeness are issues in some jurisdictions and certainly could be argued should be more so as technology makes this easier. I think factors around threat to reputation and so on also are raised in new ways with deep fakes vs remixing and adding commentary to real, existing works (which can in turn be referenced by anyone who sees the parody). Even if there is a disclaimer on the original deep fake, as a de novo work which itself might get spread around without context it's at least different then what we've had until now.