It's rather the opposite. When people take a cab, they don't take a car, and won't spend time driving around looking for a parking space.
Having enough taxicabs also reduces the amount of drink driving, which is a serious problem here in the US.
And you free up parking lots and parking garages, which can be used for other infrastructure, which reduces the need to travel even more.
I've lived in cities with plenty of taxis, and I've lived in cities with next to none. The cities that had a surplus of taxis also had the least amount of traffic problems.
London has around twice as many taxicabs as New York City, for a comparable population size. Other European cities have an even higher ratio of taxis per citizens, with a 1:100 ratio not being uncommon. And those cities have the least amount of problems with automobile traffic too.