Yes, there is a vineyard in the center of Paris. Vignes du Clos is likely the smallest vineyard in the entire world and it is right in Montmartre. The vineyard is not owned by a rich family or a business. It is owned by the 18th arrondissement which is referred to as Montmartre. Vignes du Clos is basically a park that produces local wine and uses the proceeds for social initiatives.
Another beautiful thing to bring the local Parisian community together around something so sacred as wine.