Hi,
The way you describe what is going on it sounds like it’s the Thermocouple or Flame Sensor, you need to take a look to see which one you have fitted to your furnace.
They both look very similar and they both sense the pilot fame, but work differently. The Thermocouple is made two different metals and when heated it produces D.C. voltage from the energy of the flame. If it starts to break down it can take longer to get heated or stop working all together.
First you can try cleaning the probe, it that doesn’t work you will need to buy a new Thermocouple or Flame Sensor.
regards from Noelene