The Mairie de Paris has published a useful list of Paris parks and gardens that allow dogs (PDF) — well worth consulting, because Paris is pretty strict about dogs in parks. [7/1/13 Please note that this link still works, but the resulting document is organized slightly differently, and more clearly, than I’ve described below.]
The name of the park is in the left column. The middle column shows that dogs are allowed on that property; if there’s a caption, it describes the limits to the area where dogs are allowed. The right-hand column describes the parts of the park where dogs are not allowed. So, for example, in the Jardins des Champs-Elysées, dogs are allowed on the “allées cavalières” or bridle paths, which I assume means the Allée Marcel Proust and similar. Dogs are not allowed in the “aires de jeux pour les enfants,” or children’s play areas.
Remember to bring a poop bag with you! Picking up after your dog is the rule, even if it’s not always observed by the locals.