Do cows digest cellulose by using parasitic cellulose digesting bacteria?

Symbiotic, not parasitic. The cow has a 4 chamber stomach, one chamber called the rumen contains these bacteria. When the cow is chewing cud, she brings up partially digested grass, re-chews it, and swallows it again into a different chamber of the stomach.

Without these symbiotic bacteria the cow could no live on grass, as the mammalian digestive system is unable to digest cellulose into glucose. The bacteria do this for the cow, while the cow gives them a safe place to live with plenty of food (much more than they need for their own use).