I just loaded Doom 2016 and tried it.
Here’s a screen snip showing the hit point of the Plasma Cannon clearly being on a line between the end of the gun and the crosshair. There’s a bit of scatter so the exact hit point varies, but it’s clearly off-center and more or less matches the percieved end of the gun.

Here’s a snip of me standing still and firing. And as far as I can tell it’s just obeying the principles of perspective drawing. The projectiles always seem to be below and right of the crosshair until they reach it.

The projectiles seem to spawn at some point a short distance in front of the gun. There are then layers of effects - recoil, camera shake, muzzle flash, smoke - which pretty effectively obfuscate the exact position… usually.
One case where it doesn’t hide the spawn position effectively is where I was experimenting with the rocket launcher, standing against a wall which was covering the right half of the screen. The gun animation pointed sideways, indicating that I was obstructed. If I fired anyway the projectile spawn point was moved significantly and obviously to the left. The same happened with the plasma gun. As a result, the first projectile fired appears to spawn off-screen to my left. Importantly, even when they do this, they still travel in an apparently straight line towards whatever the crosshair is pointed at.
When I stand near a wall that obscures the left side of the screen, the gun shows as obstructed and pointing to the left, and hit marks on the wall appear to my left.
So, as far as I can tell, their solution is to simply spawn the projectiles in front of the gun, and point them at whatever the crosshair is centered on. I can’t conclusively rule out further subtleties, such as a curve in the trajectory, but I also can’t see the need for them - there’s enough visual obfuscation going on, and the deviations are usually small anyway. Don’t make your solution more complex than it has to be.
Actually, your weapon’s muzzle isn’t really that far sideways of the center of the screen. I just measured in Doom. It’s significantly less than 300px off center at 1920x1080. And our eyes / brain are used to making sense of exactly this kind of thing, every single day.
Basically, I don’t think the parallax problem is as big as you seem to think. Start by just pointing the projectile at whatever the crosshair is centered on, then test your game. You can always revisit if it turns out to be an issue.