Right now, my regular gaming group is in the middle of a long term D&D campaign, so I don't use any parts!
But back in the day, I seriously stripped down the Classic trav rules.
Parts of Classic Traveller I Did Not Use:
Character death during chargen. After the first time, I said, "Lame!' and made it an automatic discharge. This was back in 1977-78 before that became a semi-official rule.
Positive DMs to hit for poor armor. I never liked that some armor made you an
easier target. I only used the negative DMs. Of course, this made jack and mesh pretty much useless, but made unarmored NPC civilians much more likely to survive.
Most of the computer programming rules. Why have a weapon that can't fire without a particular program? Why have a special program just so your pilot and gunner skills actually mean something? I assumed all those programs were hard wired into the equipment from the start. Only specialty programs (Auto Fire, Anti Hijack, etc.) were not.
Vectors. Not once, ever.
Most of the OTU. I used Regina subsector as the start of my first real campaign, tried using the Spinward Marches for another, then made up my own cobbled together setting. Any published adventures I used had the names, serial numbers, etc. filed off.
Vilani names. I still remember introducing the major villain in the "Sky Raiders" trilogy, a tough, Sidney Greenstreet/ Jabba the Hutt mobster, destined to hound the PCs for three exciting adventures. His name? Enerei Kalamanaru. Say it out loud: Eh-neer-ee Kal-uh-man-uh-roo. Doesn't it just say, "I'm a rough, tough gangster, don't mess with me, or you'll be sleeping with the space mantas?" No? My players didn't think so either. They laughed and laughed, and laughed some more. No more Vilani names.
Most of the expanded character books (Mercenary, High Guard, etc.). I liked the simplicity of the original rules so much. I liked that you could roll up a character in a few minutes. I hated most of the heavier weapons in Book 4 (FGMPs, PGMPs, especially). Only Citizens of the Imperium got use from me.
To me, this was the strength of the original rules. They were so simple and straightforward, you could tinker away and almost never break them (not that we didn't try :devil

.