Blinking is a motor function controlled by the facial nerve, the seventh cranial nerve. Cranial nerves come directly from the brainstem, bypassing the spinal cord. Cranial nerve reflexes are often used to assess levels of brain function (diencephalon, mesencephalon, and medulla).
People are usually paralysed because the spinal cord has been damaged. To paralyse the legs, this can happen quite low down the spine; to paralyse the arms it's occurs higher up; to paralyse from the neck down, the spine is damaged in the neck area.
The nerves that control things on the face connect directly to the brain through the scull, so they are not likely to be damaged as easily as the spinal cord.
Simple answer: the neural pathway between the brain and the eye muscles is still intact, while the pathways between say, the brain and the arms/legs/torso are not.
If you suffer a spinal injury in your lower back, you may lose function and feeling in your legs. Suffer the same injury higher up, and you may lose the ability to control your diaphragm, which would require you to be on a ventilator.