So, the weak spot of buildings in our neighborhood is often the parapet, not because parapets were poorly designed but because they were poorly maintained, and can become sources of leaks. The brown metal siding that you see on some buildings, basically encasing the parapet, was one solution, mostly done in the 1980s (along with what's called parging, basically a cement-like material applied to the brick to prevent water penetration). This approach has fallen out of favor because it frequently causes the brick under it to deteriorate. Brick, architects are fond of saying, needs to breathe.
These days there are new ways to address parapet problems -- from applying breathable sealants to rebuilding the parapet, with new flashing etc.
The flat part of the roof has different challenges, which may include the drains. Sometimes, the roof around the drains deteriorates, enables water to seep through around it, or the drains themselves rot and don't lead the water out properly. How that gets addressed depends on the specifics.
Re: the flat roof, there are now new materials much better than tar that are widely in use. I'm not sure what you mean when you say "for environmentally conscious use of a NYC roof" but one example is the light-colored reflecting roof coating that deflects light and keep the building from heating in the summer. If you mean that you're considering turning your roof into a deck, a whole new set of rules and materials come into play.
If you are considering roof replacement, you'd likely need to get an architect (these days, there is no way to do it without one) who will draw up plans, file, supervise the work etc. If there's one lesson I've learned, it's that architects can be a huge problem and expense, so you'd want to vet properly the people you're considering.
Good luck!