There are two issues with the file you provided that lead to the eyes bugging out when you parent with automatic weights.
- You have a bunch of weird keyframes on your bones.
- Your eyes have constraints that mean that their rest pose is not the same as their constrained pose.
To fix 1), it's pretty easy. Jump to some unused frame (like frame 55), enter pose mode on your armature, select all, and clear rotation, location, and scale (alt r, alt g, alt s for me), then keyframe (i for me).
To fix 2), well, if you don't want it tracking someplace weird, delete the constraints on the eye bones. Or, if you want the rest pose to be the same as the constrained pose, you can find the proper positions for your track-to targets by scaling up a duplicate of the eye bones and then snapping the cursor to the tail of these duplicates, then snapping the track target to the cursor (snap menu is shift s for me.) Then, select one of the eyebones in edit and recalculate roll to global +Z axis (recalculate roll is ctrl n for me.) With auto-mirror enabled as you have it, the other bone will also have its roll recalculated appropriately.
There are other issues remaining with the file. I would strongly recommend applying all transforms (ctrl a in object mode for me) on your armature. Your eye bones are also placed improperly-- they should be in the center of the eyeball, but they are slightly below and lateral to that point.