Only MS3s and MS4s who are doing rotations in University Hospital or the County can park in the SUNY garage across from the medical center (East 34th Street and Lenox Road).
MS1s and MS2s who want to park in the Beneficial Foundation Parking Lot between Winthrop Street and Parkside Avenue can put their names on a waiting list for a parking permit. Go to the Faculty-Student Association Office, Room 234 in the Student Center and sign up (but you need a SUNY box number). What you can do between now and getting your parking permit is pay for transient parking in the same office. I don't know what the procedure is, but you can ask the people in the same office mentioned above.
I would avoid street parking. It's inconvenient and, if it gets dark, you'd be wise not to venture to your automobile alone.
Good luck.