Parallel universes enter the conversation because we (non physicists) can't identify any scenario explainable with classical physics that allowed those paths to be taken in the allotted time and space.

Have a look at e.g. the path integral. It's a formulation for calculating the result of the mechanism behind probabilistic mechanics. I like to use the word "possible" instead of "probable", when explaining it, sometimes. When a car is hurtling down a road, it cannot really diverge from its "classical" path too much because it is already highly localized and entangled. There's a single car we're talking about. But a single isolated particle is not even so localized as to be able to be called a particle, but more so a wave of possibility of measuring in a certain state, such as a location. (I'm still a student though, so I do not yet fully understand how particle location and space-patch entanglement connect.)

But, anyway, as I grasp it so far... nature already makes it possible to look like one can "take multiple paths" when one is coherent, i.e. isolated from entanglement with environment during that path, i.e. where the "which path" is not already decided via e.g. measurement. For example, the Lamb shift is the consequence of the fact that the universe really doesn't know which paths of particle creation and annihilation, if any, the force carriers between the electron and the proton took. So, it is able to take all of them. Why should it rather be our expectation rather than how nature says it can work? It seems like a mental contortion is needed - but nature really does operate based on possibility because it's the consequence of the more fundamental fact that there being no one / nothing in the entire universe, including the particle itself, that "knows" what state it's in or going to be in, during a certain span of the system's evolution.

At the moment there is such a record keeper, the influence of "possibility" diminishes.