If all movement were parallel, how might that change the nature of time?
Einstein's theory of gravity requires that time is an integral part of space time which is 4 dimensional. True parallel motion probably does not exist in space time. Your question seems to imply a separate universe different than space time.
One question would be what "time" actually is. A time scale seems to be a measurement tool created by observers using selected "clocks" (uniformly repeating events) to measure other events by. Using that definition, I would argue that "time" is an abstract concept that has no actual reality and we should be talking about "clocks" instead. (I know this is not the majority view of what time is).
I would argue that where ever change or action of any kind exists (i.e., a change in space exists), then the concept of time exists, whether or not there are any good "clocks" available to measure time by.
I'm not sure if that addresses your question though.
What measurement (displacement, spin, mass, etc.) would have actual reality if not time?
Separate names with a comma.