There were these huge green parts on the map that I assumed were parks.
I mentioned that Tokyo had a many large parks, pointing to the green areas.
She said those are the Korean neighborhoods. The Japanese do not acknowledge the existence or publish street names in the Korean areas of Tokyo. For all intents and purposes they don't exist.
I own a large number of maps of Tokyo. What your friend told you is not true. Green areas on most maps are indeed parks or at least open areas of some sort. I opened my favorite book of maps, Tokyo 7000, and there are two areas keyed as green in it: kouen (
, parks) and ryokuchi (
, green space).
Also for what it is worth, most streets in Tokyo do not have a name. The Japanese use an area system for naming, not a street system. For instance, the address for the publisher of the map that I am using is 
3-1-2, which starts with the city, Tokyo, then the division of Tokyo called Chiyoda-ku, then to the area Misakichou San Choume within Chiyoda-ku, and then finally gives a smaller numbered area, and then the building number in that area (which turns out to appear on page 189 of their own map book.)
As to the completeness of the maps, I have walked a largish portion of Tokyo using maps such as Tokyo 7000 and I can assure you of their accuracy. Cab drivers use these highly detailed maps to find their way about the city (there is no true equivalent to London's "The Knowledge" for Tokyo cab drivers.)
(For the pictures of kanji in this posting, I used Jim Breen's excellent website.)