If time travel in both directions were possible, is the idea of a paradox a paradox?

If time travel in both directions is possible, it seems to be that there is an almost necessary implication that the timeline as a whole has “already occurred.” In other words, traveling to what is the future from our perspective would be traveling to the past from all points ahead of that, ad infinitum, so that at one point (potentially) all points in time would be in the past – i.e., the end of time.

Therefore, is a paradox possible? For instance, the idea of going into the past and killing your grandfather would mean you never existed to do it. Therefore, you couldn’t have done it. Also, in the future, your grandson could travel to his past and kill your son, and therefore would never have existed to do it.

So, if time travel is possible in both directions, does that require that the timeline be of a particular nature? (static or malleable, single or parallel, etc.) If so, what?