The importance of parallel arrows in a commutative square I noticed that whenever there is a commutative square, the relation it imposes on parallel morphisms is usually very important (e.g. natural transformations, pullbacks). In contrast, there's usually almost nothing to be said about the consecutive morphisms in these diagrams, even though the equation expressed by the square directly involves their composition. Is there a conceptual explanation for this in general?
 A: This is an interesting observation , but imho this is a case where there is less than meets the eye. Commutative squares by themselves do not really say much more about the constituent arrows - or pairs of arrows - than their commutative relation already says. In particular your examples - natural transformations and pulbacks/pushouts - derive their interest because of other/extra conditions imposed on the squares.
Take natural transformations:
the real noteworthy fact is that you can build a commutative square for all pairs of objects in the domain category. In itself a single square is not that special. It is like saying: a single puzzle tile means nothing, it is not even a rectangle, but taken together in a special way, the whole collection of tiles has the remarkable property of forming a rectangular puzzle. 
Same with pulbacks/pushouts: the real interesting thing is the the universal property.
Not convinced yet? Ok, let's dig deeper.
Take a very special commutative square: a commutative triangle.

It can be interpreted as either one of the following commutative squares with identity morphisms:


where $\text{Id}_X^{\mathbb{C}}$ stands for "the identity morphism of object X in category $\mathbf{C}$"
(Please see comment below for more info on these diagrams)
Now: in the first square, f is parallel with an identity - which is a very special morphism: iso, mono, epi... What insight can we get from this? None! f can be anything.
Same story with g in the other diagram.
Conclusion: the importance of parallel arrows in commutative squares? In general...none
A: I think you have hit upon an important idea.  We don't know the general story yet.  But my feeling is that the commutative squares (for naturality, for instance) are drawn in a particular way for a good reason, which is unfortunately not captured by the theory.  In particular, the arrows drawn horizontally play a different role from those drawn vertically.  The horizontal arrows play the role of "functions", i.e., map inputs to outputs, whereas the vertical arrows play the role of "relationships", i.e., something upstairs is related to something downstairs.  Naturality says that all "relationships" are preserved by "functions".
In Computer Science, we work with a concept called relational parametricity, which is more general than naturality, where "functions" and "relationships" are decoupled from each other.  We don't talk about "composition" of functions with relationships, but use more general axioms of fibrations to relate the two.  When "relationships" are specialized to "functions", these conditions do boil down to using composition of morphisms.  These are very intriguing facts, and it would be great for more people to get involved in investigating them.
The foundational article on relational parametricity is Reynolds: Types, abstraction and parametric polymorphism.  The categorical axiomatization I am talking about may be found in Dunphy and Reddy: Parametric Limits.
