The equation of ellipse is  .

When the ellipse crosses -axis, then .

The points are .

Slope of the tangent is derivative of the ellipse.

.

Differentiate on each side with respect to .

.

Slope of the tangent line is at : .

Slope of the tangent line at : .

Since the two slopes are equal, then the tangent at these points are parallel. 

The two slopes are equal, then the tangent at these points are parallel.