If you have power to the starter at the battery lead, and power to the solenoid lead when the key is turned to start position, the problem is the starter itself. That seems likeliest, the windings have been known to short out when hot, and I believe the temperature of the block actually rises in the moments after the motor is stopped due to the absence of coolant circulation. This heat is transferred to the starter and without air circulating around it from the car being in motion, the starter is hottest after stopping the car (unless you're idling in traffic for a while). If it's ignition switch or wiring, there'll be no voltage at the solenoid lead when key is turned to start.
-Chris