Surely that can't be true in general - I would think the quality/convenience/cost of the alternative would be by far the biggest factor (but yes, you could argue that's rolling multiple factors into one).
But as an example, plenty of offices I've worked at over the years have offered free or cheap parking to all staff, with sufficient space for virtually everyone, yet less than half of the staff chose to drive simply because getting there by car was slower/more expensive than the alternatives (typically train or tram, and a few, like myself, that much prefer our pushbikes!). Actually at my current job in a CBD/downtown building, nobody drives despite the high availability of parking, but it is quite expensive to use (~$18 a day I believe).