I ran into this problem on a 1996 Sunfire that I got for "free" from my neighbor and I wanted to let everyone know what I did to fix the problem. Some people may not want to do this, but for me it was perfectly acceptable and cost $0! 
Here is more then likely what your problem is:
MAKE SURE YOU READ THIS (it will make sense when you start taking things apart:
The parking pawl probably has a broken spring and is not working correctly OR the parking pawl itself is broken. The pawl is a 1.25" long piece of metal external to the motor and is part of the Stack-O-Plates on the output shaft of the motor and to which the wiper push rods are attached. When you turn the wipers to "Off" the motor will reverse direction and the pawl will catch on a stationary tab that is part of the motor/wiper frame. This will hold the wipers in the down position while the motor continues to run and allow the parking Stack-O-Plates-Spring-Wrapped gizmo to do its magic which drags the wiper arms down into the extra low "parked" position. If the tiny spring that holds the pawl in the out position breaks or the pawl is broken then the pawl will not engage the stationary tab. The wipers will then continue swinging upward through half a swing at which point a cam/switch that is internal to the pulse board (the cover on the motor output gear (three T20 Torx screws to remove) will tell the motor to stop because it thinks the parking Stack-O-Plates gizmo has finished doing its job but instead the wiper arms will be stuck in the "Hello, I'm a dork" raised flag position.
If you have a broken spring, you may want to fix it. Very cheap and very easy and maintains the low parking position of the wipers.
Read this archived post (which I've quoted in my post here) to fix your spring:
http://www.pontiacforum.com/pontiac/archive/index.php/t-20010.html
If you have a broken parking pawl, as I did, and you now need to replace the whole transmission assembly for $250
then read on.
This repair method will NOT allow your wipers to park in the really low parking position.But instead your wipers will park in the intermittent wiper park position. 
To see what I mean, turn your intermittent wipers on. See where they stop (park) in between swipes, that's where your wipers are going to park if you do this repair. It that's acceptable to you, read on!
If nothing else, you can do this repair to get your wipers to park in a somewhat normal position until you can afford to replace the wiper transmission.
(YOU DO NOT NEED TO REMOVE THE WIPER MOTOR TO PERFORM THIS REPAIR)
OK, now that you're all educated and hopefully have discover what your problem is, here is what you need to do.
*****MAKE SURE YOU'VE DISCONNECTED THE BATTERY BEFORE YOU PROCEED!!!*****
1) Remove the plastic covers off the wiper arm retaining nuts.
2) Remove the wiper arm nuts with a 13mm socket.
3) Twist and pull with your finger tips and finger nails at the base of the small rubber squirt tubing off the hard plastic squirt delivery tubes.
4) You may have to yard up and down on the wiper arms pretty hard to get them loose from their tapered shafts. Don't be a mindless gorilla but be brave and keep at it. They will come loose.
5) Remove the cowl cover by pulling or prying out the center of the six retaining push pins.
6) Then lift the cowl cover up then out at a 45 degree angle. You will find the cover is a very tight fit between the right hand wiper shaft and the weatherstrip at the base of the windshield. Be brave and keep pulling and prying until it comes loose.
7) Remove the three T20 Torx head bolts from the cover on the wiper motor:
8) There are two spring arm switches, the lower one uses the plastic cam mounted to the circuit board to open and close. This is what controls the intermittent park position for the intermittent wiper mode and also the initial park position when the wipers are switched off. The upper spring arm switch is used to run the motor in reverse until the cam that's on the worm gear (driven by the motor) opens the connection, thus stopping the motor.
To stop the motor from running in reverse and parking the wipers in the vertical (or "Hello, I'm a dork" raised flag) position, simply bend the UPPER spring arm so that it's always open.
9) Squirt some type of sealant around the edge of the cover so is doesn't leak.
10) Put everything back together!
NOTE:Reinstalling the cowl cover and getting the rubber weather strip over the cover is the hardest part of the whole job. Use putty knifes (slowly and carefully!) or a very small slender screwdriver. It took me half an hour just to do this part of the job that.
Your done!
Now go buy something fun with all that money saved!
Hope this helped someone!
Here is more then likely what your problem is:
MAKE SURE YOU READ THIS (it will make sense when you start taking things apart:
The parking pawl probably has a broken spring and is not working correctly OR the parking pawl itself is broken. The pawl is a 1.25" long piece of metal external to the motor and is part of the Stack-O-Plates on the output shaft of the motor and to which the wiper push rods are attached. When you turn the wipers to "Off" the motor will reverse direction and the pawl will catch on a stationary tab that is part of the motor/wiper frame. This will hold the wipers in the down position while the motor continues to run and allow the parking Stack-O-Plates-Spring-Wrapped gizmo to do its magic which drags the wiper arms down into the extra low "parked" position. If the tiny spring that holds the pawl in the out position breaks or the pawl is broken then the pawl will not engage the stationary tab. The wipers will then continue swinging upward through half a swing at which point a cam/switch that is internal to the pulse board (the cover on the motor output gear (three T20 Torx screws to remove) will tell the motor to stop because it thinks the parking Stack-O-Plates gizmo has finished doing its job but instead the wiper arms will be stuck in the "Hello, I'm a dork" raised flag position.
If you have a broken spring, you may want to fix it. Very cheap and very easy and maintains the low parking position of the wipers.
Read this archived post (which I've quoted in my post here) to fix your spring:
http://www.pontiacforum.com/pontiac/archive/index.php/t-20010.html
If you have a broken parking pawl, as I did, and you now need to replace the whole transmission assembly for $250
then read on.This repair method will NOT allow your wipers to park in the really low parking position.
But instead your wipers will park in the intermittent wiper park position. To see what I mean, turn your intermittent wipers on. See where they stop (park) in between swipes, that's where your wipers are going to park if you do this repair. It that's acceptable to you, read on!If nothing else, you can do this repair to get your wipers to park in a somewhat normal position until you can afford to replace the wiper transmission.
(YOU DO NOT NEED TO REMOVE THE WIPER MOTOR TO PERFORM THIS REPAIR)
OK, now that you're all educated and hopefully have discover what your problem is, here is what you need to do.
*****MAKE SURE YOU'VE DISCONNECTED THE BATTERY BEFORE YOU PROCEED!!!*****
1) Remove the plastic covers off the wiper arm retaining nuts.
2) Remove the wiper arm nuts with a 13mm socket.
3) Twist and pull with your finger tips and finger nails at the base of the small rubber squirt tubing off the hard plastic squirt delivery tubes.
4) You may have to yard up and down on the wiper arms pretty hard to get them loose from their tapered shafts. Don't be a mindless gorilla but be brave and keep at it. They will come loose.
5) Remove the cowl cover by pulling or prying out the center of the six retaining push pins.
6) Then lift the cowl cover up then out at a 45 degree angle. You will find the cover is a very tight fit between the right hand wiper shaft and the weatherstrip at the base of the windshield. Be brave and keep pulling and prying until it comes loose.
7) Remove the three T20 Torx head bolts from the cover on the wiper motor:
8) There are two spring arm switches, the lower one uses the plastic cam mounted to the circuit board to open and close. This is what controls the intermittent park position for the intermittent wiper mode and also the initial park position when the wipers are switched off. The upper spring arm switch is used to run the motor in reverse until the cam that's on the worm gear (driven by the motor) opens the connection, thus stopping the motor.
To stop the motor from running in reverse and parking the wipers in the vertical (or "Hello, I'm a dork" raised flag) position, simply bend the UPPER spring arm so that it's always open.
9) Squirt some type of sealant around the edge of the cover so is doesn't leak.
10) Put everything back together!
NOTE:Reinstalling the cowl cover and getting the rubber weather strip over the cover is the hardest part of the whole job. Use putty knifes (slowly and carefully!) or a very small slender screwdriver. It took me half an hour just to do this part of the job that.
Your done!
Now go buy something fun with all that money saved!
Hope this helped someone!
