When and how did the word "dimension" take on the popular connotation of "parallel universe"?

Especially in the mid-twentieth century, popular science fiction would often talk about "creatures from another dimension" and use similar language that suggested that a "dimension" refers to some kind of parallel or alternate universe. This usage is extremely different from the way that the term is used in actual mathematics, where "dimension" simply refers to the number of independent coordinates required to specify a location within an object. Loosely speaking, in the technical mathematical usage a "dimension" is more like a direction than a location. (It also doesn't really make any sense to talk about a being living "in" a particular dimension in the mathematical sense.)

How and when did "dimension" come to take on this connotation that seems completely unrelated to its actual technical meaning?

As mentioned in the comments, Science Fiction writers, led by the mathematician Charles Howard Hinton, explored the consequences of the fact that the mathematical theory of space can be extended to dimensions of number greater than the 3 we see around us. If one could "access", "manipulate" or "travel" in a fourth dimension, i.e. one independent of the three we can directly experience, you would be moving through a form of space which is exactly like what we mean by "parallel universe" - as people in the 'regular' universe would be unable to witness any changes that took place there, and yet 'real' physical interactions and experiences could happen solely in that extra dimension. A good way of accessing an intuition about this is imagine if everything that went on in a room was projected onto a wall - say an observer is outside a marquee or tent and sees the shadows on the wall. Then if people move toward or away from the wall the observer won't see (much of)* a change, so any interactions that happen in that dimension - the dimension perpendicular to the wall - happen in a world inaccessible to the observer - just like being in a 'parallel universe'.

It's fascinating that this question has come up as within the last couple of weeks I have read both a review of a novel about Hinton https://www.theguardian.com/books/2020/apr/23/hinton-by-mark-blacklock-review-voyages-into-the-fourth-dimension and an article by the author of that novel about the fourth dimesnion in science fiction https://www.theguardian.com/books/2020/jun/03/top-10-four-dimensional-novels-mark-blacklock-hinton.

• the 'projection' in this real-life example doesn't work like a perfect mathematical projection, but I hope you still see the point