The French call it le pique-nique (peek-neek).  Here are my favorite places to enjoy a picnic in Paris:

Tuileries Gardens
Métro:  Concorde or Tuileries or Palais Royal-Musée du Louvre.  Bus: 24, 72, 73, 94, 84.
The Central Park of Paris, although not as expansive.  Located in front of the Louvre on the right bank. The grounds are perfect for a delightful stroll or a picnic.

Place du Vert Galant
Métro:  Pont-Neuf or Cité.  Bus: 21, 24, 27, 38, 47, 85, 96.
This park is located on the tip of Île de la Cité near Pont-Neuf.  This is a great spot to watch the bateaux sail up and down the Seine while enjoying a baguette avec fromage and a taste of wine.  

Luxembourg Gardens
Métro:  Odéon, Cardinal Lemoine or Saint-Sulpice.  Bus: 21, 27, 38, 82, 83, 84, 89, 95.
Located in the Latin Quarter.  This sizable park is the home to royal gardens and the subject of many Impressionist paintings.  It’s the perfect place for an afternoon stroll.  Be like a real Parisian and plan “un pique nique” lunch here with cheese, baguette and obligatory bottle of wine. 

Place des Vosges
Métro:  St-Paul.   Bus: 29, 69, 76, 96.
Located in the heart of the Marais, this is the oldest public square in Paris.  It is surrounded by picturesque 17th century townhouses. Drop in on the splendid art galleries around the square.  Visit Victor Hugo’s home at 6, place de Vosges.  30-45 minutes.  Victor Hugo home website:  www.maisonsvictorhugo.paris.fr.

There is even a company in Paris that will help you arrange a picnic:  http://www.picnicsparis.com