Horizontal runs left to right like the "horizon" when you look out at a body of water such as the ocean. Parallel means side-by-side. Items are "parallel" to one another. Horizontal to each other is incorrect, it's one line.
I don't think horizontal to each other makes any sense because the "other" is not specified in the question and is thus undefined.. In parallel, it is always assumed that you are referring to 2 lines that ALWAYS maintain an equal distance from each other. In this case the "other" is always assumed to refers to a second line.