In terms of understanding the science and math, this is a good starting point for the layman.
(I'm expanding this answer on request, to make a more complete answer that addresses the different concepts in parallel universes.)
Let's start by pointing out that this essay is far from a complete resource on the topic of parallel universes and I'd encourage all who read it to supplement with their own research on the matter. I'll also point out that the very concept of whether or not a multiverse is possible or not is hotly contested between physicists across the world, even today. Finally, I'm going to point out that we don't even know all there is to know about this universe yet and it may never be possible to do so; scientists today often refer to the 'observable universe' as a concept that defines the radius of the universe (with Earth at the centre), the length of which is the distance that light can travel between the big bang and now. As such, we can never hope to know the 'complete' universe now, let alone the nature of a multiverse, but we do have a number of mathematical models and resultant theories we can explore.
Let's look at two of them; the many-worlds scenario and string theory.
The Many-Worlds Scenario is based on an idea from quantum physics that maybe really small sub-atomic particles don't work on a concept of waveform collapse, or put more simply, don't resolve into a final state from their 'superposition' before they're observed.
This concept of waveform collapse was explored by Erwin Schrodinger in his famous thought experiment about the cat that sits in the box, neither alive nor dead until 'observed', or interacted with. In the many-worlds model, the universe splits at the point that the cat is observed. In one, the cat is alive and in the other the cat is dead, leading to a virtually infinite number of universes that represent every possible 'state' of matter (and energy).
If we use this definition of the universe, then yes, we enter parallel universes all the time without even trying.
On the other hand, getting to the spirit of your question; can one accidentally move between these parallel universes so that one version of ourselves is in a universe 'intended' for a different version? No.
Again, starting with the many worlds scenario. What is not clear from my own research is where all these universes reside by comparison to each other, but let's assume for the moment that each of the many universes is in itself a fully formed 4 dimensional spacetime construct. Let's further assume that these universes are all finite in nature and exist side by side, with the closest one being the one that matches our existing situation most closely, the changes that differentiate them appearing only at the end of the spacetime construct.
To get to a universe in which you'd notice any substantial change between the two, you'd essentially have to instantly travel an (almost) virtually infinite number of universe lengths in an instant to get to the 'new' earth. This is prohibited by relativity of course, which means you need a wormhole.
The energy requirements alone for these phenomena would be prohibitive. They require negative energy to maintain, but that is likely going to require an equal amount of positive energy to establish, at the very least. To maintain a 1 cubic metre opening on one end of a wormhole, you need an energy mass of around the size of Jupiter. Passing through something that needs that much energy to maintain will be noticed, and very dangerous. It's probably not surviveable. As for accidental, well the probability of landing on a parallel earth in another 'dimension', with each dimension containing an observable universe the size of ours? Well, it's as close to mathematically impossible as makes no odds.
But, what about string theory?
Well, the parallel universes that arise from string theory math are even more exotic. Here we come across a concept called 'branes', short for membranes, of lower dimensionality that are tied to the 9-dimensionality of strings.
I won't go into explaining this in detail because it's not relevant to my answer, other than to say that this theory states that the complete mulitverse can be made up of many different branes of differing dimenionality, all comprised within a 9-dimensional construct of which quantum strings are a primary component. In this model, the parallel universes are far fewer, far more exotic by comparison to each other, and can only be traversed via extremely localised points at which the branes intersect.
Again, you're looking at wormholes, with all the energy baggage they carry. BUT, this time there are more problems. Your wormhole may not lead to a universe that has the same number of dimensions, may not have environments in them (and not be capable of supporting them either) that are conducive to human life, and their location in our universe would dictate that you can only have fixed wormholes, in fixed places, connecting to fixed alternate universes.
This is actually far better for your chances of the wormhole travel being accidental, but far worse for your chances of surviving the travel or ending up somewhere surviveable.
In short, regardless of whether your multiverse is the many-worlds or string theory model, if you were to (even accidentally) travel between universes;
1) You'd notice the transit
2) You probably wouldn't survive it
3) the probability of 'landing' in a safe, surviveable place on the other side alone would make the journey so implausible as to be impossible in every practical sense.
That said, yours would not be the first story to do it anyway. :)