As the

Resistance ships are running away and the transports are being picked off one by one

the shots fired travel in what appears to be a parabolic trajectory, arcing “up”, then back “down” to hit their targets. But since in space there’s no up and down defined by a main gravitational force, why would the projectiles not simply travel in a straight line?

Cinematically the visual effect seems to be derived from something like a catapult or trebuchet. But why in the Star Wars world would a weapon behave like that? In-universe answers only, please.

The shots definitely follow an arcing path towards their target. While we can understand why this was styled as such for out-of-universe reasons, I've yet to find any rational in-universe explaination for it that matches previous canon. That said, there are at least two possible reasons I can think of.

1) They may be near a gravity well of some sort

While the planet was obviously quite far away, I don't believe we ever see from space the star it orbits. It's possible it was "down" from our perspective in space even while unseen, which could explain the need to calculate an arc trajectory to compensate for gravity. A nearby black hole is likewise a possibility.

2) The shots themselves are capable of adjusting course

Similar to proton torpedos in A New Hope, but perhaps more akin to the Starkiller Base splitting beam, we've seen non-missle weapons capable of changing course mid-flight in canon. Considering the shots almost never seemed to miss, perhaps they were able to "lock on" or be drawn to the target regardless of firing angle, and the arc was intended to hit their shields from different sides to find a weakness. This wouldn't be too far fetched, considering all the new tech the First Order seems to have access to.

I admit both of these are unsubstantiated, and without more information there's no way to be certain. But until we're given more information, these are at least potential explainations for the behavior we see.

Ambush Recap

Admiral Ackbar: All craft, full engines; concentrate rear shields.

1. The Raddus blocks capital ship fire with rear shields.


2. Kylo's TIE fighters fire missiles that blow up the Raddus' bridge.


3. The Ninka blocks capital ship fire with aft shields.


4. Kylo's TIE fighters strafe the Ninka, causing explosions on the bow.

Explanation

Clearly, to survive a close-range barrage long enough to gain distance, the Resistance vessels had to concentrate all of their shielding in the rear. This leaves the ships vulnerable even to TIE fighters, let alone a glancing blow from capital ship weapons.

I think what we're seeing is the First Order trying to hit an unshielded part of the vessels by curving their shots around the concentrated rear shields. Militaries have been known to slow or guide projectiles to get around defenses, and in this case a slower shot allows more of a curve. Furthermore the shields hug the cigar-like shape of the Raddus' hull quite closely, so using only a slight curve would allow the shots to retain most of their velocity rather than going for a widely arcing shot, which may be dodgeable.

As for why they would continue to do this when targeting unshielded transports, perhaps recalibrating the weapons for fast straight shots would have taken too long.

Discarded Explanation

I'd considered given Snoke's demonstrated mastery of telekinesis and his throne room looking glass that perhaps he was using the Force to slow and bend turbolaser bolts (a la Kylo in TFA), but alas the curving continues after his death and while Kylo is otherwise engaged.

The transports that are being shot are on their way to a Crait, and they aren't traveling at lightspeed, so the planet is pretty close. The weapons officer (or weapons droid) probably realizes this and compensates for the gravitational pull when firing.

(other answers mentioned gravity and the planet, but perhaps left out the obvious point that the planet is their actual destination - the closer they get, the stronger gravity would be)

This was simply for visual effect. IN reality, 1 the planet is not in the right angle to allow for gravity to work on the weapon. Secondly the sun would have to be really close to have such a pull on a small particle beam (laser beam) Remember, gravity has more effect the more mass you have. Third, due to lack of gas resistance and gravity, you have nothing to curve the shot. This is a laser beam, not a torpedo that could have a mechanism to change course mid flight. Beams cannot be redirected without a physical reflective unit.

Hence.. just for visual aaaah and oooo's. Science wise (not that Star wars is based on much science at all), it's still part of this universe and dimension, just in an other galaxy therefor still would fall part of the same rules of physics as we have.

I think the real answer is that there's no good reason (and honestly, it and me a lot).

However, I believe all of the space battle scenes take place near a planet. Perhaps the projectiles have some kind of solid component and are behind effected by gravity.