My go-to resource for working through this was this guide by the man Pedrogas himself:
It is aimed at solo play, although you can do it coop too.
Just be aware that the forklift glitch he shows in Nightfall does not work in coop, the enemies will spawn. And you will have more enemies in general.
Apparently it works fine in coop, but the players who aren't clipping need to be a bit far from the gate.
I had a total of about 3 hours 50 minutes after playing it the first time. After using the tips he showcases, I was able to shave off the 50 minutes pretty comfortably.
You do not
need to hit par time on every level, although it obviously helps. You have quite a bit of leeway. All the par times combined add up to 2 hours 35 minutes.
Combining solo and coop works. My first playthrough was on coop, I then improved my times with a mix of solo and coop. I had not beaten every level solo legendary when I got the achievement.
The achievement itself did NOT immediately unlock. It unlocked when I started a mission after having brought my total below 3 hours in a previous one.
You can check your current sum total by going to: Options & Career > Career > Stats > Halo Reach > Campaign, then scroll down to Fastest Clear Time (Legendary). To check individual level times, instead go to Leaderboards and view each level individually.
(I'll note that I am on PC, the above menus might be slightly different on console, I don't know)