In overcrowded cities, parking space becomes a scarce resource especially during peak hours, when citizens are looking for a vacant parking spot near their work or their children's school. The high demand for parking spaces near these locations leads to traffic jams and incidents due to drivers cruising the streets or around parking lots in search of an unoccupied space. Drivers feel even more frustrated when they are forced to give up a spot that is coveted by several other drivers at the same time, once they get there. Most of the proposed smart parking solutions do not address the problems that occur when multiple vehicles are chasing the same single space. In this context, we propose CAS-SP, a Collision-Avoidance Scheme for Smart Parking that aims to help drivers shorten their search time inside parking lots, reduce costs by consuming less fuel and feel less stressed during this daily routine task. Our scheme is inspired by the CSMA/CA method, widely deployed in wireless networks. It allows deferring the vehicle entry to the parking lot by applying a tuned backoff selection strategy. We investigate our proposed scheme under different scenarios related to the parking lot architecture (single level, multi-level), the parking lot occupancy and the number of incoming vehicles. The obtained results show the effectiveness of our scheme in resolving the contention, reducing the collision among vehicles by about 88%, shortening the wasted time in searching for a vacant place inside the parking lot and saving about 450 s for a multi-level parking lot with many entrances.