Why are equatorial bonds perfectly parallel to ring bonds?

Equatorial bonds are parallel to ring bonds because of the staggered arrangement of the bonds in a cyclohexane chair.

It may be easier to see this if you look at a Newman projection for cyclohexane.

(from img.sparknotes.com)

Let's say that the lower arrow points to `"C-1"` and `"C-2"` is to the right of `"C-1"`.

Then `"C-1"` is the carbon atom at the front of the right hand circle and `"C-2"` is the hidden carbon behind it.

The equatorial `"C-H"` bond at `"C-1"` bond is parallel to the bond between `"C-2"` and `"C-3"` and also to the bond between `"C-5"` and `"C-6"`.

I have highlighted those bonds below to highlight the parallel bonds.

(adapted from img.sparknotes.com)