Why is Paris still polluted during the Covid-19 shutdown?
There are fewer cars. Are they only a small part of Paris' air
  pollution?


Recent evolution of European standard on emissions for private vehicle means that the average car is polluting orders of magnitude less than the average gasoline-burning heater. 

Even for a city plagued by the widespread use of huge, inefficent cars (New York), the heating systems (heat and hot water) may account for >40% of the emissions https://blogs.ei.columbia.edu/2019/01/15/heat-pumps-home-heating/

A couple of emissions pie-charts for PM10 and PM2.5 for Paris in 2012 (from the report Inventaire régional des émissions en Ile-de-France Année de référence 2012 - éléments synthétiques Edition mai 2016 )


and