That's a classic IAC symptom.
The IAC is also known as the "throttle bypass valve", and it does just that: It allows air to go around the throttle (which is closed if your foot is off of the gas pedal). If the IAC sticks shut, there isn't enough air to start the car. Pushing the gas pedal opens the throttle, supplying the missing air and allowing the engine to start.
Often, a sticky IAC will start to work once the car warms up, but will stop working again when cold.
Replace it or clean it.
