Or you can directly parent an object to an armature bone by going into Pose Mode and selecting the object then shift-selecting the bone you want to parent to then Ctrl-P -> Make Parent To -> Bone . No need to add the armature modifier at all …
Or if for some reason you do want to join your various parts into a single mesh you can do that too as long as you keep the parts discreet, i.e. not connected by edges or faces . And when you go to parent to the armature (in Pose Mode) do this : Ctrl-P -> Make Parent To -> Armature -> Create Vertex Groups ? -> Name Groups .
Now when you select the compound mesh/object and look in the Links and Materials panel in the editing context (F9) in the Buttons Window you will have a named vertex group for each bone in your armature (the vertex groups are named after each armature bone name so you might want to put some thought into naming them and not just the default Bone, Bone.001, etc. as this will help a lot when setting up something complicated) .
With the named vertex groups assigning the various parts to each bone is a breeze . Just tab into Edit Mode for the mesh/object and simply select the discreet parts by hovering your cursor over it and hitting the L hotkey (to select linked meshes) then go to the Links and Materials panel and select from the list of named vertex groups you want to assign that part to and simply hit “Assign” . Now simply deselect all (A key twice) and repeat until all your mesh is assigned to a vertex group .
Obviously the first method is faster, but if you have a complex scene and don’t want a lot of clutter in the Outliner or for whatever other reason you might have you do have another option (and it also is a good way to familiarize yourself with how vertex groups work with the armature) . Never the less both methods will work the same as far as the animation is concerned .