As with any great walking city, like NYC, London or Rome, Paris has numerous places to spend some time that cost nothing and make you feel more like a resident, if you have the time and inkling to pause and enjoy them. There are also lots of places to stop and relax and just take in the scenery or do some people watching, or enjoy a coffee, a drink, a glass of wine, a pastry or even a light snack.
The mentioned Parc Monceau, just a short walk along Avenue Hoche from the Arc de Triomphe, is a favourite of ours. We discovered it quite by accident and now end up spending some time there on most visits. Lots of good suggestions in the article, but two additional favourites of ours would be Place des Voges in the Marais, not far from the Canal St-Martin, and on the Left Bank, the Luxembourg Gardens adjacent to the Luxembourg Palace near the Latin/student quarter. Both were built in the early 17th century.
Place des Voges is the oldest planned square in Paris, a lovely oasis surrounded by stunning architecture. It has had many famous residents over the last 400 years, not the least of whom was Victor Hugo of "Les Misérables" fame. The Luxembourg Gardens, like the mentioned Champs de Mars and the Tuileries Gardens, is a wonderful place to just sit and enjoy the fountains, the statuary and the formal gardens themselves. Of course, a walk along the banks of the Seine is another wonderful way to pass some time for free and take in the city and its history.