Car says I'm moving when I'm not?

My girlfriend just got a 98 Plymouth Neon and the previous owners had it sitting up for around a year with a dead battery. It was given to us as no charge so I couldn't turn it down. After replacing the battery, car started right up. I noticed, however, sometimes when I start it and run it, the speedometer shows it to be going between 15 and 20 mph and the odometer changes as if it were moving, but it is in park sitting in the driveway. After it begins to warm up a litte it stops and the speedometer drops back to 0. Previous owner doesn't know anything about it. Anybody know what could cause this to happen?