What parts of Quito doesnt have any tourists?

Wow you better be careful - Quito is well know to rival Costa del sol with being overrun with English speakers. Having a Homer brain that drains of knowledge hearing English speaking you probably should stay home & listen to CDs just to be safe...

I don't think you have to worry about other gringos. Most of them don't even acknowledge each other's presence.

Does it have to be Quito? There are language schools in other parts of Ecuador such as Otavalo or Cuenca, to name just two. Both of those places have fewer tourists most of the time than Quito does. Having said that, it´s quite easy to spend time in Quito without speaking English, even in gringolandia I speak spanish in the cafes, bars, internet places, etc. There are also quite a few Colombians and Argentinians around so you can make friends with other travellers and also improve your spanish.