I had the exact same problem with my 2000 buick century which i also bought used and after asking GM what it could possibly be they told me it was the whole wiper arm assembly which would be over $700 to replace. I decided to have a look myself and discovered that a small metal tab which makes the wipers park was bent and the mechanism was not engaging after turning the wipers off. i simply bent the tab back carefully and they parked again, saving me hundreds and it took about 15 minutes.
Open the hood and look where the wiper arms attach to the spindle. remove the caps covering the nuts, then remove the nuts. the hardest part here is most likely going to be getting those arms off. then remove the plastic grille under the right wiper arm (driver's side).
When you can get at the motor assembly, turn the wipers on and off and watch how it assembly operates and see if you can find the tab. This is the best picture i could find with google.
Once i found the tab i took a large flat head screwdriver and tapped it gently with a hammer to bend the tab back. you can tell when you turn the wipers on and off that they park properly. When it came time to put the wiper arms back onto the spindles i set them where they would be when turned off (resting in the plastic cradle at the lowest position).
I hope this was helpful. i know it was for me.