The Seine's banks (especially between Pont Neuf and Pont d'Iéna) are car-free and pedestrian-friendly. Picnic with wine is a Parisian tradition.
Buy a baguette, cheese, charcuterie, and a bottle of wine at a Monoprix or Carrefour City. Find a spot on the quay.
The islands (Île de la Cité, Île Saint-Louis) are perfect picnic islands. Berthillon ice cream on Île Saint-Louis is mandatory after dinner.