When you shut the car off, the pump is done creating vacuum. But there is still vacuum in the system (via checkvalves), that's how your locks still work for awile, ect. This vacuum sucks oil right up to the ignition switch, that's where it goes next if the shutoff valve is leaking oil. First-hand experience here.
I cleaned out the lines with brake cleaner, worked great. Make sure they are dry before reassembling. I also replaced the rubber connectors which had gotten quite soft in several places.
Ensure the shutoff valve goes in correctly (potential run-away engine!!!!!!!) if you find that to be the source of your issue.
Check the top of your radio for oil!

__________________
Think Alternative Energy!
300CD '80 (now gone but not forgotten...)