Parenting eyes moves them inwards I'm very new to blender and I've run into a problem that I don't know how to solve.

Here is the face of the character:



I recently added eyes to it and now I need them to follow the character when I animate it. 

I added two new bones and placed them in the eyes. Then I selected the eyes > selected the bones > Ctrl+P > With Automatic Weights. Here is the result:


As you can see, the eyes move inwards slightly. How do I make them stay at the same place? Thanks.
First, select your armature, switch to Pose mode and come back to its Rest position. 
Apply the Mirror modifier of your eye object.
Separate each eye with P two make 2 separate objects.
Select each eye and parent it with ctrl P to each eye bone.
It works.