Have a 1999 Scenic and the wipers have a mind of their own, sometimes won't stop, sometimes stop half way through sweep very random and downright dangerous as they stop at anytime.
Any help appreciated.
Your Scenic should still have an Interconnection Unit. This is similar to a UCH but with out the immobilser function. This is seperate.
Its likely to be either the switch (as has already been mentioned) in the motor or the wiring. What engine do ou have. If its a Diesel, i've had the loom rub through on the intercooler causing the wipers not to park. The wipers will park after a set period of time if no park message is sensed.
Get the circuit checked by a competent electrician, preferably a Renault dealership!
I have the exact same model (similar age) and suffer the same problem - intermittently, it only happens when it is cold and damp.
After a while it goes away. But I would be very interested in knowing the solution as it is very annoying when it happens.
The self-parking mechanism on many wiper motors rely on a having a good clean earth. Check this out first - on a multimeter it should read 0 ohms. A poor earth connection may vary greatly depending on the weather - the wetter it is the more likely it is to be a good connection but poor when dry. Not saying it'll cure the prob but worth trying first.