Yes — your skepticism is basically right.
There really is an interesting scientific question lurking underneath these stories: humans appear to have undergone a fairly dramatic change in behavior somewhere in the broad neighborhood of 50,000–100,000 years ago. We see increasingly elaborate symbolic behavior, art, personal ornaments, long-distance exchange, sophisticated tools, and eventually very rapid cultural accumulation. But the old idea of a single sudden “Great Leap Forward” around exactly 50,000 years ago is itself much less secure than popular articles often suggest. Evidence for symbolic and complex behavior appears considerably earlier and develops unevenly.
Where things go off the rails is the next step:
“We don't know exactly why this happened, therefore perhaps consciousness became connected to a parallel universe.”
That doesn't explain the phenomenon. It merely relocates the unexplained part.
And there's an especially important distinction here. A hypothesis such as “some interaction with another universe caused the change” would need to tell us what interaction, by what physical mechanism, under what conditions, and what observable consequences it produces. If it can't do that, “parallel universe” is functioning rather like “the gods did it” with more sophisticated vocabulary.
There are legitimate multiverse ideas in physics, but they don't generally provide a mechanism by which human brains suddenly acquire an additional channel of consciousness. The many-worlds interpretation of quantum mechanics, for example, isn't a theory of psychic communication between universes.
The thing that caught my eye in searching is that this particular claim seems to have quite a lot of New Age/UFO/consciousness literature attached to it. One example explicitly describes a supposed worldwide “consciousness boost” around 50,000 years ago and attributes it to access to a new spectrum of consciousness. That's very much the sort of thing I'd put in the interesting speculation, spectacularly short on mechanism category.
And I think your phrase “an explanation that doesn't explain, just pushes the question away” is exactly the right criticism. It's essentially:
Why did human cognition change?
→ Because of a connection to another universe.
Why should that connection change cognition?
→ [new mystery]
You've just moved the question one level farther away.
The genuinely interesting question, in my view, is why human cultural evolution seems to have accelerated so dramatically, and whether that required a genetic change, changes in population structure and social learning, language, cumulative culture, ecological circumstances, or simply a combination of things. That's a real scientific puzzle. We don't need to import another universe to make it puzzling.
Yes, with one qualification: it weakens the case for a sudden origin of consciousness, but it doesn't rule out a specific event affecting specifically human consciousness or cognition.
I think your distinction is an important one. If by consciousness we mean something like having subjective experiences of the world, pleasure and pain, fear, memory, attention, and dreams, then there is no obvious reason to put its origin at 50,000 years ago. Mammals clearly exhibit many of those capacities, and birds and some other animals have surprisingly sophisticated cognitive and behavioral abilities. Dreaming is particularly suggestive because it implies internally generated experience rather than merely responding mechanically to the immediate environment.
What seems much more distinctive about Homo sapiens is not consciousness itself but the combination of language, abstract representation, exceptionally flexible intelligence, and cumulative culture. An animal can have an experience without being able to describe it; it can remember without constructing a written history; it can solve a problem without transmitting the solution through generations in a way that permits others to improve upon it.
So I'd separate two questions:
“When did consciousness appear?” — probably a very ancient evolutionary development, with no reason to expect a particular date.
“When did humans acquire our extraordinary capacity for cumulative symbolic culture?” — that's the genuinely interesting question, and roughly 50,000 years ago may be part of its history, although the archaeological record suggests a much more gradual and complicated development.
There is also a nice logical point in what you're saying. If somebody observes a sudden increase in the sophistication of human behavior and concludes that consciousness itself suddenly appeared, they're already making an assumption that intelligence, language, culture and subjective experience are essentially the same thing. They aren't.
And dreams make the argument particularly awkward for the “sudden consciousness” story. A sleeping dog doesn't have language or human-style abstract reasoning, but it can apparently have an internally generated experiential state. Whatever consciousness ultimately turns out to be, evolution almost certainly didn't wait around until one particular human generation and then switch it on.
So I'd say your formulation is stronger than simply “animals are conscious.” If consciousness is an evolved biological phenomenon, its evolutionary history should substantially predate the remarkable cultural explosion of our species. The latter needs explaining; it doesn't automatically require explaining the former.