No on both for reef. Parrot eats algae from rocks and is known to damage corals in the process of getting to the algae that's on or in the rocks. Maybe other than the blue jaw, no other trigger is reef safe as they love all those inverts.
Not really sure about this...
As long as you stay with the Scarus genus of parrotfish, you are good. Many on here have quoyi and (like myself) princess parrotfish. They do spend a good amount of time picking at the rock, but are very careful not to pick at any coral. They pick around it very well, and I have never seen mine damage any coral at all, and I have a mixed reef with both lps, sps, and a very tempting brain on my sandbed. When food lands on and around my coral, the parrotfish simply won't go after it. She'll swim up to it, then swim away looking for easier pickings...
Often times, many people will get the parrotfish we keep in our reefs confused with this:
http://www.youtube.com/watch?v=8T-gnZevlaY
Completely different type of parrotfish...
As for triggers, most Xanthichthys are safe. I had a pair in my 225g along with the aforementioned parrot for a couple of years with absolutely no issues at all in regards to coral or inverts. (My male trigger forced my female into starvation with his over exuberant mating attempts, and she passed last week.) I had all 3 fish with just about every invert you can imagine (from nudibranchs, to decorative clams, to shrimp, to crabs) with absolutely no issues at all. I will advise that triggers are very inquisitive fish, though, and will try to pick up anything they can and move it around. If it is glued down, they tend to leave it alone, so if you glue down your coral, you are good...
Experiences with parrots and triggers: They are very active and large fish in comparison to most frags, so from time to time, I have witnessed one of my fish swim past a sps frag, and accidentally break it when they ran into it as they swam by, but my foxface has done the same. This will happen with larger fish...
Also: My male trigger and parrotfish fought quite a bit when the male trigger was attempting to mate with my female trigger, so if you do the combination, I'd suggest not to have them all three together with an established trigger pair. After the female trigger passed, the male trigger and parrotfish stopped fighting...
In conclusion, the scarus parrots offered for trade in this hobby and the Xanthichthys triggers are as reef safe as a large fish can be. They may knock over your frags, but won't eat your coral or inverts. But each individual fish may vary (as always)...