After my 35th time failing to get through Tip of the Spear on Heroic with 17 skulls on, I decided enough was enough. I did some modifying of my own and testing things, and I think I have the best solution, that works even better than some of the other guides in here.
As others have mentioned, the par score is heavily reliant on the par time. As you pass the par time, your score multiplier of 3x slowly drops to 1x, and then after enough time, gives you no points for kills. This is why it's essential to beat the level in par time, or at least very close to it. I had a few levels where I was about 1 minute over par, and still managed to get it ok.
The first thing I'll say is that you need to be familiar with the level. Run it through a few times on normal difficulty to see where dangerous enemies are. Fuel Rod Grunts, Energy Sword Elites, Gravity Hammer Brutes... Find them and be on the lookout when you see them, because not only does dying decrease your score, but the clock keeps ticking, and some of those checkpoints are kinda wacky.
You're going to want to play through the levels on Normal Difficulty, with Tilt, Famine, Cloud, Catch, and Thunderstorm. A lot of the other guides suggest Heroic, but I am not that good at the game. If you are good enough, feel free to bump it up to Heroic and turn off Catch or Tilt or whatever you'd like to make it a little easier.
Now here's the catch. All the other guides have how to get scoring, and how it works, but I'm going to tell you the secret to getting through the levels quickly and easily: DMR and Plasma Pistol.
I may not be the first to say this, and if someone else has mentioned it in a video or another guide, I do apologize. But the Plasma Pistol, especially with Tilt, is arguably the best weapon in the game. Tilt makes Elites weaker to plasma weapons and stronger to assault weapons. Spraying them with 3 clips of an assault rifle barely breaks their shield (and with Catch on, they've likely stuck you in that time), but 10-15 shots from a Plasma Pistol takes it down incredibly easily. If you're close enough, you can melee them, otherwise zoom in and headshot with the DMR. I will mention that you should be rapid firing the pistol, never charge it as it takes too much energy to do so and if it misses, you're out of luck. Only use the charge shot if you need to disable a vehicle.
This works on other enemies as well. Grunts? 4 or so plasma shots and they're down. (It takes 2 headshots from a DMR to kill the special ones otherwise.) Jackals? Just hit them and they go down eventually, and a plus is they usually drop more Plasma Pistols. Surely Brutes take more than a handful of plasma shots to die, right? Wrong. Just keep hitting them and they go down faster than Elites. Head doesn't even matter, they take flat damage so as long as you keep blasting them they'll die. Hunters are trickier, but dodge and weave their slam attacks and spray them in the back, and they will go down as well, sooner or later. The only thing you really will want a DMR for is headshotting the Elites and then taking out the Engineers that you see occasionally.
With your DMR and Plasma Pistol, make your way through the levels, trying to grab more ammo from Spartan drop points when you can or grabbing new pistols. It also helps immensely to figure out what the beginning of the different points are (Exodus, for example, has 3: Alpha is running through the city and waiting for the elevator, Bravo is jetpacking across the sky tram station, and Charlie is turning on the missile launchers). This lets you gauge time and figure out how ahead or behind you are. Some missions, like The Package, have incredibly uneven sections, though, so it will take a bit of working with it to figure out how much time you need for each.
If you're finding yourself having trouble managing to make the score even in the time for the level, you can always turn on Iron, which gives you a nice score bonus. The only downside is that if you die, you have to restart the level. For 10-minute par time missions, this isn't awful, but it's still incredibly frustrating and I'd first look and see if there's any high priority enemies you can destroy in the mission first.
For vehicle missions, it's usually best to use the vehicle given to you. Rocket Warthog on Tip of the Spear and Scorpion on The Package will do much more damage and get you many more points than a plasma pistol ever will.
Hopefully this helps people who are struggling to figure out how to make it through the missions, either in time or by getting past the hard parts. I personally had a lot of trouble with getting through missions on Heroic without dying or passing time, so I figured I would just share the solution I finally figured out.
Let me know if you have any questions or if this is still not working for you. I also recommend you check out the individual achievements you're looking for, as they have specialized guides for those specific levels that may help you more than a blanket loadout.