TL;DNR: No, but if it's not various things are harder to do, especially if you have multiple objects controlled by the armature.
Short answer: No, it doesn't matter, so long as one is parented to the other. However parenting the object to the armature is common, and animators will expect to be able to move both by moving the armature. If you parent the armature to the object instead, then to move both at once you have to remember to move the object, rather than the armature. It's even more of a problem for an animator if the armature is used to control multiple objects.
It is a matter of workflow, that is, how things are done. As you become more experienced using an application like blender you tend to learn sets of steps for doing similar things. Adding a rig to a mesh is one of those things and there's a pretty efficient workflow that most people adopt variations of:
- Create the mesh
- Create and position the armature
- Parent the mesh to the armature, using "with automatic weights", or, perhaps "with empty groups"
- Use weight painting (and other techniques) to adjust the way the armature moves the mesh.
The usual way of doing step 3 results in the armature being the parent of the mesh. A lot of other tools assume that relationship, so by not doing it you may break things.
When you parent with automatic weights, Blender does more than just make the armature the parent of the mesh:
- It makes the armature the parent of the mesh object
- It adds an armature modifier to the mesh object
- It creates vertex groups for the mesh object, one per deforming bone in the armature. (A bone is "deforming" if moving it is meant to deform the mesh. Armatures often have bones that are used to control other bones and are not considered deforming.)
- It attempts to fill out the vertex groups with vertices that should move when the related bone moves, and gives them "weight" -- a measure of how much influence the bone's movement should have on the vertices.
You can do all of these by hand. You can even omit the first step and not parent either to the other. The problem with not parenting is that you have to remember to move both at once if you want them to stay together. The problem with parenting the armature to the mesh is that you run contrary to "usual" practice, and that can be confusing.
If you use your armature to control multiple objects, then not parenting the objects to the mesh makes moving everything together very complicated.
So it's most common for the armature to be the parent of the mesh.