In dreams it's common for houses and/or cars to represent the self. So, my guess is that this is what's going on here. The fact that you're falling asleep may mean that you've gotten into a pattern or entrenched routine in your life - so much so that you're either beginning to lose interest, or you're ignoring common obstacles around you. This may mean that you're purposely turning a blind eye, or (as you mentioned), you're not seeing them until it's too late. Interestingly, this is a perfect representation of a "road block" in terms of something stopping you from where you want or need to be in life.
That said, dreams are also all about how we feel. So, how are you feeling in this dream (other than sleepy)? Are you frustrated, scared, or annoyed? What is the emotion that's going on when you're trying to park and hitting other cars? Also, is there a feeling when you turn off onto the normal street (relief, sadness, etc)?
This dream is referring to your spiritual state and where you are going in your personal life. If you are falling asleep at the wheel then it is showing there is something in your life that is interfering in what direction you want to take. Swerving in two different lanes is showing a personal decision you are trying to make in your life. With you backing into a parking spot and hitting cars around you is showing how you are having a hard time staying with this choice in your life as it may interfere with other's feelings relating to this choice that you may not be able to see or are aware of. With going about 5 mph is showing how you are taking your time on making this decision.