Parasite? You mean predator. Parasites generally specialize in their food source.
Cows, horses, and deer eat all the vegetation in sight and move on. Doesn't make them parasites.
Omnivore is merely a term of convenience, and applies to most mammals if taken literally - almost any can eat both meat and plant/fungus/algae. Humans simply tend to insist on combining both plant and animal consumption in a day, often at every meal, due to availability of both, so "omnivore" is most convenient.