Learn French: Nous avons marché à travers la forêt.

Because we are moving across a continuous physical space. Parmi means "among" and is used for being surrounded by distinct, countable items in a group. If you wanted to use parmi, you would say parmi les arbres (among the trees), but for passing through a single continuous environment like la forêt, à travers is the correct choice.

Could I use par instead of à travers to mean "through" in this sentence?

Generally, no. À travers specifically emphasizes the physical action of crossing a space or obstacle from one side to the other. Par is used more broadly for routes (like passing "by" or "via" a place) or for looking through openings, such as par la fenêtre (through the window).

Since marcher is a verb of movement, shouldn't it use être in the passé composé instead of avoir?

This is a classic refresher! Not all verbs of movement use être. The verbs that take être are usually those indicating a change of location, direction, or state (like aller, partir, arriver). Verbs that describe the manner of movement, such as marcher (to walk), courir (to run), or nager (to swim), take avoir.