I don't think alternate universes are parallel. If you think about it, what occurs in nature that is parallel? Nothing. Only people try to make things straight and tidy.
Alternate universes and dimensions probably exist in random, wavy lines, not straight, parallel ones. And they probably intersect at points along themselves, thus creating nexus points where two or more universes touch.
At these nexus points, it is probable that some version of you could slip into our universe, and you could slip into theirs. Possibly without either of you even realizing it.
Sliding into different versions of their universes, where maybe the only difference is that you like the colour red and the other version of you likes blue. No one would notice, not even you, until you started to slowly realize everything you own had blue hues instead of red....
Unless it was a Renaissance universe. Then you'd probably notice right away. What with the tights and fancy wigs.