I don't think you need to know the number of commuting cars. As you just mentioned some cars are commuting, it means they may need a temporary parking space. In the latest trends of and studies by planners and designers, it is highly suggested to decrease the number of cars in towns and give the priority to public transit , bikers and pedestrians. The reason is as long as you provide wider streets and more parking spaces, you are making the city more car-friendly in oppose to people-friendly. A livable city is city designed for people not for cars.
In case you still to know how to figure out the number of cars in a city, the best place to ask is normally the transport division of municipality or traffic police. They should have the counts of cars and other types of data regarding different types of vehicles, peak hours and ....
What you need is DIRECTIONAL traffic survey. Typically PROFILE traffic surveys are done (by road authorities), but these provide you numbers only, not origins and destinations.
As for as designing the parking is concerned you need not have the exact counting of the vehicles/cars commuting the city rather the parking requirement may drastically vary based on the activity. There may be a number of cars in the CBD or any other area which may not require the parking at all - (pick and drop facility); some may require the temporary parking or short term parking, some vehicles may also need a long term parking ranging up to few hours which may not be permanent as in case someone came for meeting, function, and shopping, etc., but at the same time a large number of may also require a regular or periodic permanent parking.
There may be a co-relation between the number of registered vehicles/cars in the city, and the parking requirement in an activity zone or CBD but there are many factors like origin-destination of the trips, concentration of the activity, the number of CBDs, hierarchy of activity centres which may affect the parking requirement.
As for as the data related to the number of cars is concerned it may be obtained from the vehicles registration and licencing agency and/or transportation offices/division of the municipal government.
Please see the highway capacity manual 2010 Vol 3 Chap 16. It will help you to describe the traffic situation in your city (with the existing parking maneuvers, the number of intersections, pedestrians, etc..) subsequently improving the mobility by changing the different parameters and performing some previous calculus