When you burn oil, you take carbon from the ground and introduce it into the atmosphere. This increases the total amount of carbon resident in the atmosphere.
When you grow plants, you take carbon out of the atmosphere. When you eat it, you release it back into the atmosphere. The total amount of carbon resident in the atmosphere does not change.
Grass grows more vigorously when regularly pruned, for example by a grazing animal. Farms also make a point of planting thing which grow quickly and efficiently. Thus it's reasonable to expect that farming could increase the rate of carbon turnover in the atmosphere, which would create the appearance of increased carbon emissions without a corresponding increase in the amount of carbon resident in the atmosphere.
In practice, farms regularly use synthetic fertilizers. They might cut down trees to clear fields for animals. They use heavy, oil burning equipment. These, and many other practices, actually increase atmospheric carbon, and for that reason should be discouraged. But the cow itself only causes a short lived, fixed increase in atmospheric methane and an increased rate of carbon turnover.