You can parent an object to a specific bone using a 'Child Of' constraint:

Here, the object I want to parent to an individual bone has a 'Child Of' constraint added, with the armature and bone I want it to follow selected from the 'Target' and 'Bone' search boxes.

Pressing the 'Set Inverse' button may be required to return the object to it's original position as after choosing the target object it may jump to that bone's position.

If you want a bone to have a parent bone you have to do the parenting while in 'Edit Mode'. At the minute you are in 'Object Mode'. From the header of the 3D view, with the armature selected, choose 'Edit Mode':

(The bone you want to be the parent will also have to be in the same armature for it to be available as the parent).

If you want to delete, let say, bone1, and you want to keep, let say, bone2, with influence of bone1 being given to bone2, you can delete bone1, then go in the Properties panel > Object Data > Vertex Groups, select bone1 group, press Select, select bone2 group, press Assign, now the vertex group that was linked to bone1 is linked to bone2.

You can select the bone you want parented, go to Properties > Bone settings > relations > Parent, and set the bone you want as a parent there.