I propose that in Tallinn are many car parkings near Old city. But there is one good what I can suggest which is very close to everything. It is located in Vabaduse valjak - the square which is near St John's church. It maybe 5 minutes walk to Kiek in de Kok.
When you're leaving parking you need to pay directly, there are no parking-machines.
As you know Estonia is going to join the European Union in 2004. Seems the first step is already done: in Tallinn prices are as high as in most other European metropolises.
If you see this sign you will be charged for parking. You shouldn't forget unless you have enough money. I saw a lot of traffic wardens in the streets!