Hi Sandy,
As rents increase, residents are forced to live more densely, so we have a similar problem at my building.
We have about 150 residents but only 103 parking spaces in our lot. About 90 of the spaces are assigned. We are in a business area, so there is no residential street parking nearby. We require applicants to provide vehicle information before an application is accepted, and if someone has three cars, we have to tell them they can't park them all here.
I've spent 5 years trying to market to prospects who work within walking distance or use public transportation, and telling every prospect that we have only ONE parking space per apartment. It takes time, but with consistency, the number of residents you have with vehicles will decline.
You might have to consider assigned or permit parking. You could also designate a specific guest parking area and prohibit residents from using it. A little work up front will pay big dividends later.
Good Luck!