I have a 1987 Plymouth Voyager with a EFI V6 and an automatic transmission.
When the temperature is 32 degrees fahrenheit or below I have to hold the key in the start position for 60 to 75 seconds before the starter engages and turns over the engine. Once it finally does the engine will start in 4 turns or so.
Below 15 degrees fahrenheit I have to hold the key in the start position for 3 or more minutes before anything happens. Again once the starter finally starts moving the engine starts in only a few turns.
The van only has 177,000 miles on it, the engine is in excellent condition (no noises, smoke or oil consumption), and the battery is 4 months old.
Above 32 degrees fahrenheit the starter engages right away with no delay. Up to about 20 minutes after shutdown on a cold day I can start the van right up. After that it gets cold and there is a delay. An hour after shutdown and I have to hold the key in the start position for a while to get it to start again.
My guess is something in the car freezes and by holding the key in the start position I am thawing it by application of electricity until whatever is frozen has thawed sufficiently to allow the starter to engage.
What could this be?
There may be a faulty starter solenoid. These are built into the starter these days.
Ensure the battery has a FULL charge and both battery cables are in good condition and the connections are clean and wrench tight at BOTH ends.
You might have a flaky neutral safety switch. Try wiggling the gear selector while the key is in the start position. Also try starting in neutral instead of park. This is a long shot, but worth a try.
You could also try tapping the starter with a piece of wood (2 X 4?)and see if it help get the starter going.